-- Module: Mathlib.Algebra.Order.Hom.Monoid
Goal:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst : Preorder α
inst_1 : Preorder β
inst_2 : Preorder γ
inst_3 : Preorder δ
inst_4 : Mul α
inst_5 : Mul β
inst_6 : Mul γ
inst_7 : Mul δ
f g_1 g : α ≃*o β
toFun : α → β
invFun : β → α
left_inv : LeftInverse invFun toFun
right_inv : RightInverse invFun toFun
map_mul' :
  ∀ (x y : α),
    { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv : α ≃ β }.toFun (x * y) =
      { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv : α ≃ β }.toFun x *
        { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv : α ≃ β }.toFun y
map_le_map_iff' :
  ∀ {a b : α},
    { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul' :
              α ≃* β }.toFun
          a ≤
        { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul' :
              α ≃* β }.toFun
          b ↔
      a ≤ b
h₁ :
  { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
        map_le_map_iff' := @map_le_map_iff' : α ≃*o β }.toFun =
    g.toFun
h₂ :
  { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
        map_le_map_iff' := @map_le_map_iff' : α ≃*o β }.invFun =
    g.invFun
obtain :
  ∀ (toFun_1 : α → β) (invFun_1 : β → α) (left_inv_1 : LeftInverse invFun_1 toFun_1)
    (right_inv_1 : RightInverse invFun_1 toFun_1)
    (map_mul'_1 :
      ∀ (x y : α),
        { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1 : α ≃ β }.toFun
            (x * y) =
          { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1 : α ≃ β }.toFun x *
            { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1 : α ≃ β }.toFun y)
    (map_le_map_iff'_1 :
      ∀ {a b : α},
        { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1,
                  map_mul' := map_mul'_1 : α ≃* β }.toFun
              a ≤
            { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1,
                  map_mul' := map_mul'_1 : α ≃* β }.toFun
              b ↔
          a ≤ b)
    (h₁ :
      { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
            map_le_map_iff' := @map_le_map_iff' : α ≃*o β }.toFun =
        { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1,
            map_mul' := map_mul'_1, map_le_map_iff' := @map_le_map_iff'_1 : α ≃*o β }.toFun)
    (h₂ :
      { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
            map_le_map_iff' := @map_le_map_iff' : α ≃*o β }.invFun =
        { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1,
            map_mul' := map_mul'_1, map_le_map_iff' := @map_le_map_iff'_1 : α ≃*o β }.invFun),
    { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
        map_le_map_iff' := @map_le_map_iff' : α ≃*o β } =
      { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1, map_mul' := map_mul'_1,
        map_le_map_iff' := @map_le_map_iff'_1 }
⊢ { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
      map_le_map_iff' := @map_le_map_iff' : α ≃*o β } =
    g

Tactic:
obtain ⟨⟨⟨_, _⟩⟩, _⟩ := g