(* Proof of lem_rImg_eq_rImg2                                              *)
intros; apply axs_extensionnalite; unfold iff in |- *; split; intros.
unfold rImg in H.
elim (axs_comprehension (fun y : E => exists x : E, In (couple x y) r) b v2);
intros; generalize (H0 H); clear H H0 H1; intros.
elim H; clear H; intros; elim H0; clear H0; intros.
unfold rImg2 in |- *.
elim
(axs_comprehension (fun y : E => exists x : E, In (couple x y) r)
(reunion (reunion r)) v2); intros; apply H2; clear H1 H2;
intros.
split; [ idtac | exists x; auto with zfc ].
elim (axs_reunion (reunion r) v2); intros; apply H2; clear H1 H2; intros.
exists (paire x v2); split;
[ idtac | elim (axs_paire x v2 v2); intros; apply H2; right; reflexivity ].
elim (axs_reunion r (paire x v2)); intros; apply H2; clear H1 H2; intros.
exists (couple x v2); split;
[ auto with zfc
| unfold couple in |- *;
elim (axs_paire (singleton x) (paire x v2) (paire x v2));
intros; apply H2; right; reflexivity ].
unfold rImg2 in H.
elim
(axs_comprehension (fun y : E => exists x : E, In (couple x y) r)
(reunion (reunion r)) v2); intros; generalize (H0 H);
clear H H0 H1; intros.
elim H; clear H; intros; elim H0; clear H0; intros.
unfold rImg in |- *.
elim (axs_comprehension (fun y : E => exists x : E, In (couple x y) r) b v2);
intros; apply H2; clear H1 H2; intros.
split; [ idtac | exists x; auto with zfc ].
unfold rel_sig in rRel; unfold inc in rRel.
generalize (rRel (couple x v2) H0); clear rRel H H0; intros.
elim (lem_cartesian_propertie a b (couple x v2)); intros; generalize (H0 H);
clear H H0 H1; intros.
elim H; clear H; intros; elim H; clear H; intros; elim H; clear H; intros;
elim H0; clear H0; intros.
elim (lem_couple_propertie x x0 v2 x1); intros; generalize (H2 H1);
clear H1 H2 H3; intros.
elim H1; clear H1; intros; generalize H0; rewrite H2; tauto.
Qed.
(* End of proof of lem_rImg_eq_rImg2                                       *)

(***************************************************************************)
(* Si (x,y) est dans r alors x est dans le domaine de r.                   *)
(***************************************************************************)
Lemma lem_rel_and_rdom :
forall (r a b : E) (rRel : rel_sig r a b) (x y : E),
In (couple x y) r -> In x (rdom r a b rRel).