/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov

! This file was ported from Lean 3 source module order.semiconj_Sup
! leanprover-community/mathlib commit c3291da49cfa65f0d43b094750541c0731edc932
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Logic.Function.Conjugate
import Mathbin.Order.Bounds.OrderIso
import Mathbin.Order.ConditionallyCompleteLattice.Basic
import Mathbin.Order.RelIso.Group
import Mathbin.Order.OrdContinuous
import Mathbin.Algebra.Hom.Equiv.Units.Basic

/-!
# Semiconjugate by `Sup`

> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.

In this file we prove two facts about semiconjugate (families of) functions.

First, if an order isomorphism `fa : α → α` is semiconjugate to an order embedding `fb : β → β` by
`g : α → β`, then `fb` is semiconjugate to `fa` by `y ↦ Sup {x | g x ≤ y}`, see
`semiconj.symm_adjoint`.

Second, consider two actions `f₁ f₂ : G → α → α` of a group on a complete lattice by order
isomorphisms. Then the map `x ↦ ⨆ g : G, (f₁ g)⁻¹ (f₂ g x)` semiconjugates each `f₁ g'` to `f₂ g'`,
see `function.Sup_div_semiconj`.  In the case of a conditionally complete lattice, a similar
statement holds true under an additional assumption that each set `{(f₁ g)⁻¹ (f₂ g x) | g : G}` is
bounded above, see `function.cSup_div_semiconj`.

The lemmas come from [Étienne Ghys, Groupes d'homeomorphismes du cercle et cohomologie
bornee][ghys87:groupes], Proposition 2.1 and 5.4 respectively. In the paper they are formulated for
homeomorphisms of the circle, so in order to apply results from this file one has to lift these
homeomorphisms to the real line first.
-/

variable {α β γ : Type _}

open Set

#print IsOrderRightAdjoint /-
/-- We say that `g : β → α` is an order right adjoint function for `f : α → β` if it sends each `y`
to a least upper bound for `{x | f x ≤ y}`. If `α` is a partial order, and `f : α → β` has
a right adjoint, then this right adjoint is unique. -/
def IsOrderRightAdjoint [Preorder α] [Preorder β] (f : α → β) (g : β → α) :=
  ∀ y, IsLUB { x | f x ≤ y } (g y)
#align is_order_right_adjoint IsOrderRightAdjoint
-/

/- warning: is_order_right_adjoint_Sup -> isOrderRightAdjoint_supₛ is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : Preorder.{u2} β] (f : α -> β), IsOrderRightAdjoint.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) _inst_2 f (fun (y : β) => SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (x : α) => LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2) (f x) y)))
but is expected to have type
  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : CompleteLattice.{u2} α] [_inst_2 : Preorder.{u1} β] (f : α -> β), IsOrderRightAdjoint.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))) _inst_2 f (fun (y : β) => SupSet.supₛ.{u2} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (setOf.{u2} α (fun (x : α) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) (f x) y)))
Case conversion may be inaccurate. Consider using '#align is_order_right_adjoint_Sup isOrderRightAdjoint_supₛₓ'. -/
theorem isOrderRightAdjoint_supₛ [CompleteLattice α] [Preorder β] (f : α → β) :
    IsOrderRightAdjoint f fun y => supₛ { x | f x ≤ y } := fun y => isLUB_supₛ _
#align is_order_right_adjoint_Sup isOrderRightAdjoint_supₛ

/- warning: is_order_right_adjoint_cSup -> isOrderRightAdjoint_csupₛ is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLattice.{u1} α] [_inst_2 : Preorder.{u2} β] (f : α -> β), (forall (y : β), Exists.{succ u1} α (fun (x : α) => LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2) (f x) y)) -> (forall (y : β), BddAbove.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))) (setOf.{u1} α (fun (x : α) => LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2) (f x) y))) -> (IsOrderRightAdjoint.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))) _inst_2 f (fun (y : β) => SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α _inst_1) (setOf.{u1} α (fun (x : α) => LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2) (f x) y))))
but is expected to have type
  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLattice.{u2} α] [_inst_2 : Preorder.{u1} β] (f : α -> β), (forall (y : β), Exists.{succ u2} α (fun (x : α) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) (f x) y)) -> (forall (y : β), BddAbove.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))) (setOf.{u2} α (fun (x : α) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) (f x) y))) -> (IsOrderRightAdjoint.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))) _inst_2 f (fun (y : β) => SupSet.supₛ.{u2} α (ConditionallyCompleteLattice.toSupSet.{u2} α _inst_1) (setOf.{u2} α (fun (x : α) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) (f x) y))))
Case conversion may be inaccurate. Consider using '#align is_order_right_adjoint_cSup isOrderRightAdjoint_csupₛₓ'. -/
theorem isOrderRightAdjoint_csupₛ [ConditionallyCompleteLattice α] [Preorder β] (f : α → β)
    (hne : ∀ y, ∃ x, f x ≤ y) (hbdd : ∀ y, BddAbove { x | f x ≤ y }) :
    IsOrderRightAdjoint f fun y => supₛ { x | f x ≤ y } := fun y => isLUB_csupₛ (hne y) (hbdd y)
#align is_order_right_adjoint_cSup isOrderRightAdjoint_csupₛ

namespace IsOrderRightAdjoint

/- warning: is_order_right_adjoint.unique -> IsOrderRightAdjoint.unique is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PartialOrder.{u1} α] [_inst_2 : Preorder.{u2} β] {f : α -> β} {g₁ : β -> α} {g₂ : β -> α}, (IsOrderRightAdjoint.{u1, u2} α β (PartialOrder.toPreorder.{u1} α _inst_1) _inst_2 f g₁) -> (IsOrderRightAdjoint.{u1, u2} α β (PartialOrder.toPreorder.{u1} α _inst_1) _inst_2 f g₂) -> (Eq.{max (succ u2) (succ u1)} (β -> α) g₁ g₂)
but is expected to have type
  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : PartialOrder.{u2} α] [_inst_2 : Preorder.{u1} β] {f : α -> β} {g₁ : β -> α} {g₂ : β -> α}, (IsOrderRightAdjoint.{u2, u1} α β (PartialOrder.toPreorder.{u2} α _inst_1) _inst_2 f g₁) -> (IsOrderRightAdjoint.{u2, u1} α β (PartialOrder.toPreorder.{u2} α _inst_1) _inst_2 f g₂) -> (Eq.{max (succ u2) (succ u1)} (β -> α) g₁ g₂)
Case conversion may be inaccurate. Consider using '#align is_order_right_adjoint.unique IsOrderRightAdjoint.uniqueₓ'. -/
protected theorem unique [PartialOrder α] [Preorder β] {f : α → β} {g₁ g₂ : β → α}
    (h₁ : IsOrderRightAdjoint f g₁) (h₂ : IsOrderRightAdjoint f g₂) : g₁ = g₂ :=
  funext fun y => (h₁ y).unique (h₂ y)
#align is_order_right_adjoint.unique IsOrderRightAdjoint.unique

/- warning: is_order_right_adjoint.right_mono -> IsOrderRightAdjoint.right_mono is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Preorder.{u1} α] [_inst_2 : Preorder.{u2} β] {f : α -> β} {g : β -> α}, (IsOrderRightAdjoint.{u1, u2} α β _inst_1 _inst_2 f g) -> (Monotone.{u2, u1} β α _inst_2 _inst_1 g)
but is expected to have type
  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : Preorder.{u2} α] [_inst_2 : Preorder.{u1} β] {f : α -> β} {g : β -> α}, (IsOrderRightAdjoint.{u2, u1} α β _inst_1 _inst_2 f g) -> (Monotone.{u1, u2} β α _inst_2 _inst_1 g)
Case conversion may be inaccurate. Consider using '#align is_order_right_adjoint.right_mono IsOrderRightAdjoint.right_monoₓ'. -/
theorem right_mono [Preorder α] [Preorder β] {f : α → β} {g : β → α} (h : IsOrderRightAdjoint f g) :
    Monotone g := fun y₁ y₂ hy => (h y₁).mono (h y₂) fun x hx => le_trans hx hy
#align is_order_right_adjoint.right_mono IsOrderRightAdjoint.right_mono

/- warning: is_order_right_adjoint.order_iso_comp -> IsOrderRightAdjoint.orderIso_comp is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : Preorder.{u1} α] [_inst_2 : Preorder.{u2} β] [_inst_3 : Preorder.{u3} γ] {f : α -> β} {g : β -> α}, (IsOrderRightAdjoint.{u1, u2} α β _inst_1 _inst_2 f g) -> (forall (e : OrderIso.{u2, u3} β γ (Preorder.toLE.{u2} β _inst_2) (Preorder.toLE.{u3} γ _inst_3)), IsOrderRightAdjoint.{u1, u3} α γ _inst_1 _inst_3 (Function.comp.{succ u1, succ u2, succ u3} α β γ (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (OrderIso.{u2, u3} β γ (Preorder.toLE.{u2} β _inst_2) (Preorder.toLE.{u3} γ _inst_3)) (fun (_x : RelIso.{u2, u3} β γ (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.{u3} γ (Preorder.toLE.{u3} γ _inst_3))) => β -> γ) (RelIso.hasCoeToFun.{u2, u3} β γ (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.{u3} γ (Preorder.toLE.{u3} γ _inst_3))) e) f) (Function.comp.{succ u3, succ u2, succ u1} γ β α g (coeFn.{max (succ u3) (succ u2), max (succ u3) (succ u2)} (OrderIso.{u3, u2} γ β (Preorder.toLE.{u3} γ _inst_3) (Preorder.toLE.{u2} β _inst_2)) (fun (_x : RelIso.{u3, u2} γ β (LE.le.{u3} γ (Preorder.toLE.{u3} γ _inst_3)) (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2))) => γ -> β) (RelIso.hasCoeToFun.{u3, u2} γ β (LE.le.{u3} γ (Preorder.toLE.{u3} γ _inst_3)) (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2))) (OrderIso.symm.{u2, u3} β γ (Preorder.toLE.{u2} β _inst_2) (Preorder.toLE.{u3} γ _inst_3) e))))
but is expected to have type
  forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : Preorder.{u3} α] [_inst_2 : Preorder.{u2} β] [_inst_3 : Preorder.{u1} γ] {f : α -> β} {g : β -> α}, (IsOrderRightAdjoint.{u3, u2} α β _inst_1 _inst_2 f g) -> (forall (e : OrderIso.{u2, u1} β γ (Preorder.toLE.{u2} β _inst_2) (Preorder.toLE.{u1} γ _inst_3)), IsOrderRightAdjoint.{u3, u1} α γ _inst_1 _inst_3 (Function.comp.{succ u3, succ u2, succ u1} α β γ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} β γ) β (fun (_x : β) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : β) => γ) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} β γ) β γ (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} β γ)) (RelEmbedding.toEmbedding.{u2, u1} β γ (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : β) => LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : γ) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : γ) => LE.le.{u1} γ (Preorder.toLE.{u1} γ _inst_3) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} β γ (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : β) => LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : γ) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : γ) => LE.le.{u1} γ (Preorder.toLE.{u1} γ _inst_3) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) f) (Function.comp.{succ u1, succ u2, succ u3} γ β α g (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} γ β) γ (fun (_x : γ) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : γ) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} γ β) γ β (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} γ β)) (RelEmbedding.toEmbedding.{u1, u2} γ β (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : γ) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : γ) => LE.le.{u1} γ (Preorder.toLE.{u1} γ _inst_3) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : β) => LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u2} γ β (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : γ) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : γ) => LE.le.{u1} γ (Preorder.toLE.{u1} γ _inst_3) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : β) => LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u2, u1} β γ (Preorder.toLE.{u2} β _inst_2) (Preorder.toLE.{u1} γ _inst_3) e))))))
Case conversion may be inaccurate. Consider using '#align is_order_right_adjoint.order_iso_comp IsOrderRightAdjoint.orderIso_compₓ'. -/
theorem orderIso_comp [Preorder α] [Preorder β] [Preorder γ] {f : α → β} {g : β → α}
    (h : IsOrderRightAdjoint f g) (e : β ≃o γ) : IsOrderRightAdjoint (e ∘ f) (g ∘ e.symm) :=
  fun y => by simpa [e.le_symm_apply] using h (e.symm y)
#align is_order_right_adjoint.order_iso_comp IsOrderRightAdjoint.orderIso_comp

/- warning: is_order_right_adjoint.comp_order_iso -> IsOrderRightAdjoint.comp_orderIso is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : Preorder.{u1} α] [_inst_2 : Preorder.{u2} β] [_inst_3 : Preorder.{u3} γ] {f : α -> β} {g : β -> α}, (IsOrderRightAdjoint.{u1, u2} α β _inst_1 _inst_2 f g) -> (forall (e : OrderIso.{u3, u1} γ α (Preorder.toLE.{u3} γ _inst_3) (Preorder.toLE.{u1} α _inst_1)), IsOrderRightAdjoint.{u3, u2} γ β _inst_3 _inst_2 (Function.comp.{succ u3, succ u1, succ u2} γ α β f (coeFn.{max (succ u3) (succ u1), max (succ u3) (succ u1)} (OrderIso.{u3, u1} γ α (Preorder.toLE.{u3} γ _inst_3) (Preorder.toLE.{u1} α _inst_1)) (fun (_x : RelIso.{u3, u1} γ α (LE.le.{u3} γ (Preorder.toLE.{u3} γ _inst_3)) (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1))) => γ -> α) (RelIso.hasCoeToFun.{u3, u1} γ α (LE.le.{u3} γ (Preorder.toLE.{u3} γ _inst_3)) (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1))) e)) (Function.comp.{succ u2, succ u1, succ u3} β α γ (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (OrderIso.{u1, u3} α γ (Preorder.toLE.{u1} α _inst_1) (Preorder.toLE.{u3} γ _inst_3)) (fun (_x : RelIso.{u1, u3} α γ (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.{u3} γ (Preorder.toLE.{u3} γ _inst_3))) => α -> γ) (RelIso.hasCoeToFun.{u1, u3} α γ (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.{u3} γ (Preorder.toLE.{u3} γ _inst_3))) (OrderIso.symm.{u3, u1} γ α (Preorder.toLE.{u3} γ _inst_3) (Preorder.toLE.{u1} α _inst_1) e)) g))
but is expected to have type
  forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : Preorder.{u3} α] [_inst_2 : Preorder.{u2} β] [_inst_3 : Preorder.{u1} γ] {f : α -> β} {g : β -> α}, (IsOrderRightAdjoint.{u3, u2} α β _inst_1 _inst_2 f g) -> (forall (e : OrderIso.{u1, u3} γ α (Preorder.toLE.{u1} γ _inst_3) (Preorder.toLE.{u3} α _inst_1)), IsOrderRightAdjoint.{u1, u2} γ β _inst_3 _inst_2 (Function.comp.{succ u1, succ u3, succ u2} γ α β f (FunLike.coe.{max (succ u1) (succ u3), succ u1, succ u3} (Function.Embedding.{succ u1, succ u3} γ α) γ (fun (_x : γ) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : γ) => α) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u3), succ u1, succ u3} (Function.Embedding.{succ u1, succ u3} γ α) γ α (Function.instEmbeddingLikeEmbedding.{succ u1, succ u3} γ α)) (RelEmbedding.toEmbedding.{u1, u3} γ α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : γ) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : γ) => LE.le.{u1} γ (Preorder.toLE.{u1} γ _inst_3) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u3} α (Preorder.toLE.{u3} α _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u3} γ α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : γ) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : γ) => LE.le.{u1} γ (Preorder.toLE.{u1} γ _inst_3) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u3} α (Preorder.toLE.{u3} α _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)))) (Function.comp.{succ u2, succ u3, succ u1} β α γ (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (Function.Embedding.{succ u3, succ u1} α γ) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => γ) _x) (EmbeddingLike.toFunLike.{max (succ u3) (succ u1), succ u3, succ u1} (Function.Embedding.{succ u3, succ u1} α γ) α γ (Function.instEmbeddingLikeEmbedding.{succ u3, succ u1} α γ)) (RelEmbedding.toEmbedding.{u3, u1} α γ (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u3} α (Preorder.toLE.{u3} α _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : γ) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : γ) => LE.le.{u1} γ (Preorder.toLE.{u1} γ _inst_3) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u3, u1} α γ (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u3} α (Preorder.toLE.{u3} α _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : γ) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : γ) => LE.le.{u1} γ (Preorder.toLE.{u1} γ _inst_3) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u1, u3} γ α (Preorder.toLE.{u1} γ _inst_3) (Preorder.toLE.{u3} α _inst_1) e)))) g))
Case conversion may be inaccurate. Consider using '#align is_order_right_adjoint.comp_order_iso IsOrderRightAdjoint.comp_orderIsoₓ'. -/
theorem comp_orderIso [Preorder α] [Preorder β] [Preorder γ] {f : α → β} {g : β → α}
    (h : IsOrderRightAdjoint f g) (e : γ ≃o α) : IsOrderRightAdjoint (f ∘ e) (e.symm ∘ g) :=
  by
  intro y
  change IsLUB (e ⁻¹' { x | f x ≤ y }) (e.symm (g y))
  rw [e.is_lub_preimage, e.apply_symm_apply]
  exact h y
#align is_order_right_adjoint.comp_order_iso IsOrderRightAdjoint.comp_orderIso

end IsOrderRightAdjoint

namespace Function

/- warning: function.semiconj.symm_adjoint -> Function.Semiconj.symm_adjoint is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PartialOrder.{u1} α] [_inst_2 : Preorder.{u2} β] {fa : OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))} {fb : OrderEmbedding.{u2, u2} β β (Preorder.toLE.{u2} β _inst_2) (Preorder.toLE.{u2} β _inst_2)} {g : α -> β}, (Function.Semiconj.{u1, u2} α β g (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) fa) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} β β (Preorder.toLE.{u2} β _inst_2) (Preorder.toLE.{u2} β _inst_2)) (fun (_x : RelEmbedding.{u2, u2} β β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2))) => β -> β) (RelEmbedding.hasCoeToFun.{u2, u2} β β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2))) fb)) -> (forall {g' : β -> α}, (IsOrderRightAdjoint.{u1, u2} α β (PartialOrder.toPreorder.{u1} α _inst_1) _inst_2 g g') -> (Function.Semiconj.{u2, u1} β α g' (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} β β (Preorder.toLE.{u2} β _inst_2) (Preorder.toLE.{u2} β _inst_2)) (fun (_x : RelEmbedding.{u2, u2} β β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2))) => β -> β) (RelEmbedding.hasCoeToFun.{u2, u2} β β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2))) fb) (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) fa)))
but is expected to have type
  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : PartialOrder.{u2} α] [_inst_2 : Preorder.{u1} β] {fa : OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))} {fb : OrderEmbedding.{u1, u1} β β (Preorder.toLE.{u1} β _inst_2) (Preorder.toLE.{u1} β _inst_2)} {g : α -> β}, (Function.Semiconj.{u2, u1} α β g (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) fa))) (FunLike.coe.{succ u1, succ u1, succ u1} (Function.Embedding.{succ u1, succ u1} β β) β (fun (_x : β) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : β) => β) _x) (EmbeddingLike.toFunLike.{succ u1, succ u1, succ u1} (Function.Embedding.{succ u1, succ u1} β β) β β (Function.instEmbeddingLikeEmbedding.{succ u1, succ u1} β β)) (RelEmbedding.toEmbedding.{u1, u1} β β (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) fb))) -> (forall {g' : β -> α}, (IsOrderRightAdjoint.{u2, u1} α β (PartialOrder.toPreorder.{u2} α _inst_1) _inst_2 g g') -> (Function.Semiconj.{u1, u2} β α g' (FunLike.coe.{succ u1, succ u1, succ u1} (Function.Embedding.{succ u1, succ u1} β β) β (fun (_x : β) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : β) => β) _x) (EmbeddingLike.toFunLike.{succ u1, succ u1, succ u1} (Function.Embedding.{succ u1, succ u1} β β) β β (Function.instEmbeddingLikeEmbedding.{succ u1, succ u1} β β)) (RelEmbedding.toEmbedding.{u1, u1} β β (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) fb)) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) fa)))))
Case conversion may be inaccurate. Consider using '#align function.semiconj.symm_adjoint Function.Semiconj.symm_adjointₓ'. -/
/-- If an order automorphism `fa` is semiconjugate to an order embedding `fb` by a function `g`
and `g'` is an order right adjoint of `g` (i.e. `g' y = Sup {x | f x ≤ y}`), then `fb` is
semiconjugate to `fa` by `g'`.

This is a version of Proposition 2.1 from [Étienne Ghys, Groupes d'homeomorphismes du cercle et
cohomologie bornee][ghys87:groupes]. -/
theorem Semiconj.symm_adjoint [PartialOrder α] [Preorder β] {fa : α ≃o α} {fb : β ↪o β} {g : α → β}
    (h : Function.Semiconj g fa fb) {g' : β → α} (hg' : IsOrderRightAdjoint g g') :
    Function.Semiconj g' fb fa :=
  by
  refine' fun y => (hg' _).unique _
  rw [← fa.surjective.image_preimage { x | g x ≤ fb y }, preimage_set_of_eq]
  simp only [h.eq, fb.le_iff_le, fa.left_ord_continuous (hg' _)]
#align function.semiconj.symm_adjoint Function.Semiconj.symm_adjoint

variable {G : Type _}

/- warning: function.semiconj_of_is_lub -> Function.semiconj_of_isLUB is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {G : Type.{u2}} [_inst_1 : PartialOrder.{u1} α] [_inst_2 : Group.{u2} G] (f₁ : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) (f₂ : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) {h : α -> α}, (forall (x : α), IsLUB.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1) (Set.range.{u1, succ u2} α G (fun (g' : G) => coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) (Inv.inv.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toHasInv.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) f₁ g')) (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) f₂ g') x))) (h x)) -> (forall (g : G), Function.Semiconj.{u1, u1} α α h (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) f₂ g)) (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_1)))))))) f₁ g)))
but is expected to have type
  forall {α : Type.{u2}} {G : Type.{u1}} [_inst_1 : PartialOrder.{u2} α] [_inst_2 : Group.{u1} G] (f₁ : MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (f₂ : MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) {h : α -> α}, (forall (x : α), IsLUB.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1) (Set.range.{u2, succ u1} α G (fun (g' : G) => FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (Inv.inv.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) g') (InvOneClass.toInv.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) g') (DivInvOneMonoid.toInvOneClass.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) g') (DivisionMonoid.toDivInvOneMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) g') (Group.toDivisionMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) g') (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₁ g')))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₂ g'))) x))) (h x)) -> (forall (g : G), Function.Semiconj.{u2, u2} α α h (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₂ g)))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_1)) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₁ g)))))
Case conversion may be inaccurate. Consider using '#align function.semiconj_of_is_lub Function.semiconj_of_isLUBₓ'. -/
theorem semiconj_of_isLUB [PartialOrder α] [Group G] (f₁ f₂ : G →* α ≃o α) {h : α → α}
    (H : ∀ x, IsLUB (range fun g' => (f₁ g')⁻¹ (f₂ g' x)) (h x)) (g : G) :
    Function.Semiconj h (f₂ g) (f₁ g) :=
  by
  refine' fun y => (H _).unique _
  have := (f₁ g).LeftOrdContinuous (H y)
  rw [← range_comp, ← (Equiv.mulRight g).Surjective.range_comp _] at this
  simpa [(· ∘ ·)] using this
#align function.semiconj_of_is_lub Function.semiconj_of_isLUB

/- warning: function.Sup_div_semiconj -> Function.supₛ_div_semiconj is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {G : Type.{u2}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : Group.{u2} G] (f₁ : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) (f₂ : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) (g : G), Function.Semiconj.{u1, u1} α α (fun (x : α) => supᵢ.{u1, succ u2} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) G (fun (g' : G) => coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) (Inv.inv.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toHasInv.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) f₁ g')) (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) f₂ g') x))) (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) f₂ g)) (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))))))) f₁ g))
but is expected to have type
  forall {α : Type.{u2}} {G : Type.{u1}} [_inst_1 : CompleteLattice.{u2} α] [_inst_2 : Group.{u1} G] (f₁ : MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (f₂ : MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (g : G), Function.Semiconj.{u2, u2} α α (fun (x : α) => supᵢ.{u2, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) G (fun (g' : G) => FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (Inv.inv.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) g') (InvOneClass.toInv.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) g') (DivInvOneMonoid.toInvOneClass.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) g') (DivisionMonoid.toDivInvOneMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) g') (Group.toDivisionMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) g') (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₁ g')))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₂ g'))) x))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₂ g)))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₁ g))))
Case conversion may be inaccurate. Consider using '#align function.Sup_div_semiconj Function.supₛ_div_semiconjₓ'. -/
/-- Consider two actions `f₁ f₂ : G → α → α` of a group on a complete lattice by order
isomorphisms. Then the map `x ↦ ⨆ g : G, (f₁ g)⁻¹ (f₂ g x)` semiconjugates each `f₁ g'` to `f₂ g'`.

This is a version of Proposition 5.4 from [Étienne Ghys, Groupes d'homeomorphismes du cercle et
cohomologie bornee][ghys87:groupes]. -/
theorem supₛ_div_semiconj [CompleteLattice α] [Group G] (f₁ f₂ : G →* α ≃o α) (g : G) :
    Function.Semiconj (fun x => ⨆ g' : G, (f₁ g')⁻¹ (f₂ g' x)) (f₂ g) (f₁ g) :=
  semiconj_of_isLUB f₁ f₂ (fun x => isLUB_supᵢ) _
#align function.Sup_div_semiconj Function.supₛ_div_semiconj

/- warning: function.cSup_div_semiconj -> Function.csupₛ_div_semiconj is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {G : Type.{u2}} [_inst_1 : ConditionallyCompleteLattice.{u1} α] [_inst_2 : Group.{u2} G] (f₁ : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) (f₂ : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))), (forall (x : α), BddAbove.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))) (Set.range.{u1, succ u2} α G (fun (g : G) => coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) (Inv.inv.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toHasInv.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) f₁ g)) (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) f₂ g) x)))) -> (forall (g : G), Function.Semiconj.{u1, u1} α α (fun (x : α) => supᵢ.{u1, succ u2} α (ConditionallyCompleteLattice.toHasSup.{u1} α _inst_1) G (fun (g' : G) => coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) (Inv.inv.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toHasInv.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) f₁ g')) (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) f₂ g') x))) (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) f₂ g)) (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (fun (_x : RelIso.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) => α -> α) (RelIso.hasCoeToFun.{u1, u1} α α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) (coeFn.{max (succ u1) (succ u2), max (succ u2) (succ u1)} (MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) (fun (_x : MonoidHom.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) => G -> (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))) (MonoidHom.hasCoeToFun.{u2, u1} G (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_2))) (Monoid.toMulOneClass.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (DivInvMonoid.toMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (Group.toDivInvMonoid.{u1} (OrderIso.{u1, u1} α α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))))) (RelIso.group.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))))))))) f₁ g)))
but is expected to have type
  forall {α : Type.{u2}} {G : Type.{u1}} [_inst_1 : ConditionallyCompleteLattice.{u2} α] [_inst_2 : Group.{u1} G] (f₁ : MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (f₂ : MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))), (forall (x : α), BddAbove.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))) (Set.range.{u2, succ u1} α G (fun (g : G) => FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (Inv.inv.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) g) (InvOneClass.toInv.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) g) (DivInvOneMonoid.toInvOneClass.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) g) (DivisionMonoid.toDivInvOneMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) g) (Group.toDivisionMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) g) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₁ g)))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₂ g))) x)))) -> (forall (g : G), Function.Semiconj.{u2, u2} α α (fun (x : α) => supᵢ.{u2, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α _inst_1) G (fun (g' : G) => FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (Inv.inv.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) g') (InvOneClass.toInv.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) g') (DivInvOneMonoid.toInvOneClass.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) g') (DivisionMonoid.toDivInvOneMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) g') (Group.toDivisionMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) g') (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₁ g')))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₂ g'))) x))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₂ g)))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => α) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} α α) α α (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} α α)) (RelEmbedding.toEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u2} α α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (fun (a : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) a) (MulHomClass.toFunLike.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) (MonoidHomClass.toMulHomClass.{max u2 u1, u1, u2} (MonoidHom.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))) G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283))))) (MonoidHom.monoidHomClass.{u1, u2} G (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (DivInvMonoid.toMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (Group.toDivInvMonoid.{u2} (OrderIso.{u2, u2} α α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1)))))) (RelIso.instGroupRelIso.{u2} α (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α _inst_1))))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283)))))))) f₁ g)))))
Case conversion may be inaccurate. Consider using '#align function.cSup_div_semiconj Function.csupₛ_div_semiconjₓ'. -/
/-- Consider two actions `f₁ f₂ : G → α → α` of a group on a conditionally complete lattice by order
isomorphisms. Suppose that each set $s(x)=\{f_1(g)^{-1} (f_2(g)(x)) | g \in G\}$ is bounded above.
Then the map `x ↦ Sup s(x)` semiconjugates each `f₁ g'` to `f₂ g'`.

This is a version of Proposition 5.4 from [Étienne Ghys, Groupes d'homeomorphismes du cercle et
cohomologie bornee][ghys87:groupes]. -/
theorem csupₛ_div_semiconj [ConditionallyCompleteLattice α] [Group G] (f₁ f₂ : G →* α ≃o α)
    (hbdd : ∀ x, BddAbove (range fun g => (f₁ g)⁻¹ (f₂ g x))) (g : G) :
    Function.Semiconj (fun x => ⨆ g' : G, (f₁ g')⁻¹ (f₂ g' x)) (f₂ g) (f₁ g) :=
  semiconj_of_isLUB f₁ f₂ (fun x => isLUB_csupₛ (range_nonempty _) (hbdd x)) _
#align function.cSup_div_semiconj Function.csupₛ_div_semiconj

end Function

-- Guard against import creep
assert_not_exists Finset