(* Proof of lem_rel_prod_rel_sig                                           *)
unfold rel_sig in |- *; unfold inc in |- *; unfold rel_prod in |- *; intros.
elim
(axs_comprehension
(fun c : E =>
exists x : E,
(exists y : E,
(exists z : E,
In (couple x y) r1 /\ In (couple y z) r2 /\ c = couple x z)))
(cartesien a a) v0); intros; generalize (H0 H);
clear H H0 H1; intros.
elim H; clear H; intros; auto with zfc.
Qed.
(* End of proof of lem_rel_prod_rel_sig                                    *)

(* Specifications: *)
(***************************************************************************)
(* Le domaine de r1.r2 est inclus dans le domaine de r1                    *)
(***************************************************************************)
Lemma lem_rel_prod_dom :
forall (r1 r2 a : E) (rRel1 : rel_sig r1 a a) (rRel2 : rel_sig r2 a a),
inc
(rdom (rel_prod r1 r2 a rRel1 rRel2) a a
(lem_rel_prod_rel_sig r1 r2 a rRel1 rRel2)) (rdom r1 a a rRel1).