(* Proof of lem_couple_propertie_rl                                        *)
intros; elim (lem_couple_propertie al ar bl br); intros; exact (H1 H).
Qed.
(* End of proof of lem_couple_propertie_rl                                 *)
(***************************************************************************)
(* Produit cartesien de deux ensembles                                     *)
(***************************************************************************)
Definition cartesien (a b : E) :=
subset
(fun u : E =>
exists x : E, (exists y : E, In x a /\ In y b /\ u = couple x y))
(parties (parties (union a b))).

(* Specifications: *)
Lemma lem_cartesian_propertie :
forall a b x : E,
In x (cartesien a b) <->
(exists v1 : E, (exists v2 : E, In v1 a /\ In v2 b /\ x = couple v1 v2)).