(** We show that h, which is f on J and g elsewhere, is a bijection *)

    Inductive h (x y : U) : Prop :=
      | hl_intro : In U J x -> f x y -> h x y
      | hr_intro : Setminus U B (Imf J) y -> g y x -> h x y.(** We show that h, which is f on J and g elsewhere, is a bijection *)

    Inductive h (x y : U) : Prop :=
      | hl_intro : In U J x -> f x y -> h x y
      | hr_intro : Setminus U B (Imf J) y -> g y x -> h x y.(** We show that h, which is f on J and g elsewhere, is a bijection *)

    Inductive h (x y : U) : Prop :=
      | hl_intro : In U J x -> f x y -> h x y
      | hr_intro : Setminus U B (Imf J) y -> g y x -> h x y.

(* Specifications: *)
Theorem h_bij : bijection U A B h.
Lemma h1 : Rel U A B h.
Lemma h2 : to_at_most_one_output U h.
Lemma h3 : to_at_least_one_output U A h.
Lemma h4 : from_at_most_one_input U h.
Lemma h5 : from_at_least_one_input U B h.
Theorem Schroeder : A <=_card B -> B <=_card A -> A =_card B.