(* Proof of lem_Vide_Img                                                   *)
elim
(axs_comprehension (fun y : E => exists x : E, In (couple x y) Vide) a v0);
intros; generalize (H0 H); clear H H0 H1; intros;
elim H; clear H; intros; elim H0; clear H0; intros.
generalize (lem_vide_propertie (couple x v0) var_vide); intros;
absurd (In (couple x v0) Vide); auto with zfc.
Qed.
(* End of proof of lem_Vide_Img                                            *)
(***************************************************************************)
(* La relation pleine                                                      *)
(***************************************************************************)
Definition rfull (a : E) :=
subset (fun c : E => exists x : E, (exists y : E, c = couple x y))
(cartesien a a).

(* Specifications: *)
(***************************************************************************)
(* rfull est une relation sur a                                            *)
(***************************************************************************)
Lemma lem_rfull_rel_sig : forall a : E, rel_sig (rfull a) a a.
unfold rel_sig in |- *; unfold inc in |- *; unfold rfull in |- *; intros.