(* Proof of lem_rfull_Img                                                  *)
unfold rImg in |- *; intros; apply axs_extensionnalite; unfold iff in |- *;
split; intros.
elim
(axs_comprehension (fun y : E => exists x : E, In (couple x y) (rfull a)) a
v2); intros; generalize (H0 H); clear H H0 H1;
intros; elim H; clear H; intros; auto with zfc.
elim
(axs_comprehension (fun y : E => exists x : E, In (couple x y) (rfull a)) a
v2); intros; apply H1; clear H0 H1; intros; split;
[ auto with zfc | unfold rfull in |- *; exists v2 ].
elim
(axs_comprehension
(fun c : E => exists x : E, (exists y : E, c = couple x y))
(cartesien a a) (couple v2 v2)); intros; apply H1;
clear H0 H1; intros.
split;
[ elim (lem_cartesian_propertie a a (couple v2 v2)); intros; apply H1;
exists v2; exists v2; split;
[ auto with zfc | split; [ auto with zfc | reflexivity ] ]
| exists v2; exists v2; reflexivity ].
Qed.
(* End of proof of lem_rfull_Img                                           *)

(***************************************************************************)
(* La relation pleine sur a n'est autre que a x a                          *)
(***************************************************************************)
Lemma lem_rfull_eq_cart : forall a : E, rfull a = cartesien a a.
unfold rfull in |- *; intros; apply axs_extensionnalite; unfold iff in |- *;
split; intros.