Set: pp::colors
  Set: pp::unicode
  Imported 'Int'
  Assumed: A
  Assumed: B
  Assumed: f
  Defined: g
  Assumed: h
  Assumed: hinv
  Assumed: Inv
  Assumed: H1
  Proved: f_eq_g
  Proved: Inj
definition g (A : Type) (f : A → A → A) (x y : A) : A := f y x
theorem f_eq_g (A : Type) (f : A → A → A) (H1 : ∀ x y : A, f x y = f y x) : f = g A f :=
    funext (λ x : A,
              funext (λ y : A,
                        let L1 : f x y = f y x := H1 x y, L2 : f y x = g A f x y := refl (g A f x y) in trans L1 L2))
theorem Inj (A B : Type)
    (h : A → B)
    (hinv : B → A)
    (Inv : ∀ x : A, hinv (h x) = x)
    (x y : A)
    (H : h x = h y) : x = y :=
    let L1 : hinv (h x) = hinv (h y) := congr2 hinv H,
        L2 : hinv (h x) = x := Inv x,
        L3 : hinv (h y) = y := Inv y,
        L4 : x = hinv (h x) := symm L2,
        L5 : x = hinv (h y) := trans L4 L1
    in trans L5 L3
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