(* Proof of lem_rfull_prop                                                 *)
unfold rfull in |- *; intros.
elim
(axs_comprehension
(fun c : E => exists x0 : E, (exists y0 : E, c = couple x0 y0))
(cartesien a a) (couple x y)); intros; apply H2;
clear H1 H2; intros.
split;
[ elim (lem_cartesian_propertie a a (couple x y)); intros; apply H2;
exists x; exists y; split;
[ auto with zfc | split; [ auto with zfc | reflexivity ] ]
| exists x; exists y; reflexivity ].
Qed.
(* End of proof of lem_rfull_prop                                          *)
(***************************************************************************)
(* La relation identite                                                    *)
(***************************************************************************)
Definition rId (a : E) :=
subset (fun c : E => exists x : E, c = couple x x) (cartesien a a).

(***************************************************************************)
(* L'identite est une relation                                             *)
(***************************************************************************)
Lemma lem_rId_rel_sig : forall a : E, rel_sig (rId a) a a.
unfold rel_sig in |- *; unfold inc in |- *; unfold rId in |- *; intros.