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 : Π x y : A, f x y = f y x) : f = g A f :=
    Abst (λ x : A,
            Abst (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 : Π 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
10