(* Proof of lem_rel_and_rImg                                               *)
unfold rImg in |- *; intros.
elim (axs_comprehension (fun y0 : E => exists x : E, In (couple x y0) r) b y);
intros; apply H1; clear H0 H1; intros.
split; [ idtac | exists x; auto with zfc ].
generalize (rRel (couple x y) H); intros.
elim (lem_cartesian_propertie a b (couple x y)); intros; generalize (H1 H0);
clear H0 H1 H2; intros.
elim H0; clear H0; intros; elim H0; clear H0; intros; elim H0; clear H0;
intros; elim H1; clear H1; intros.
elim (lem_couple_propertie x x0 y x1); intros; generalize (H3 H2);
clear H2 H3 H4; intros; elim H2; clear H2; intros.
rewrite H3; auto with zfc.
Qed.
(* End of proof of lem_rel_and_rImg                                        *)

Lemma lem_rel_and_rImg2 :
forall (r : E) (rRel : rel r) (x y : E),
In (couple x y) r -> In y (rImg2 r rRel).