(***********************************)
(********* Intersection  ***********)
(***********************************)

(******** little intersection *******)

Definition intersection (A B : Ens) := Set_Of A (fun X : Ens => IN X B).

(* Specifications: *)
Theorem intersection_th1 : forall A B : Ens, SUB (intersection A B) A.
Theorem intersection_th2 : forall A B : Ens, SUB (intersection A B) B.
Theorem intersection_th3 : forall (A B X : Ens) (i : IN X A) (j : IN X B), IN X (intersection A B).
Lemma intersection4_Out : IN X (intersection A B).
Theorem intersection_th4 : forall (A B C : Ens) (s : SUB C A) (t : SUB C B), SUB C (intersection A B).