(*
	Cardinal.
*)

Inductive cardinal : part_set E -> nat -> Prop :=
  | cardinal_empty : forall A : part_set E, Equal A (empty E) -> cardinal A 0
  | cardinal_add :
      forall (A B : part_set E) (n : nat),
      cardinal B n ->
      forall x : E,
      ~ in_part x B -> Equal A (add_part B x) -> cardinal A (S n).(*
	Cardinal.
*)

Inductive cardinal : part_set E -> nat -> Prop :=
  | cardinal_empty : forall A : part_set E, Equal A (empty E) -> cardinal A 0
  | cardinal_add :
      forall (A B : part_set E) (n : nat),
      cardinal B n ->
      forall x : E,
      ~ in_part x B -> Equal A (add_part B x) -> cardinal A (S n).(*
	Cardinal.
*)

Inductive cardinal : part_set E -> nat -> Prop :=
  | cardinal_empty : forall A : part_set E, Equal A (empty E) -> cardinal A 0
  | cardinal_add :
      forall (A B : part_set E) (n : nat),
      cardinal B n ->
      forall x : E,
      ~ in_part x B -> Equal A (add_part B x) -> cardinal A (S n).

Lemma cardinal_sup3 : forall (A B C : part_set E) (x y : E), Equal A (add_part B x) -> Equal A (add_part C y) -> ~ in_part x B -> ~ in_part y C -> ~ Equal x y -> exists D : part_set E, Equal B (add_part D y) /\ Equal C (add_part D x) /\ ~ in_part x D /\ ~ in_part y D.