(* Proof of lem_refl_dom_inc_img                                           *)
unfold inc in |- *; intros.
unfold reflexivity in H.
generalize (H v0 H0); clear H; intros.
unfold rImg in |- *.
elim (axs_comprehension (fun y : E => exists x : E, In (couple x y) r) a v0);
intros; apply H2; clear H1 H2; intros.
split; [ idtac | exists v0; auto with zfc ].
unfold rdom in H0.
elim (axs_comprehension (fun x : E => exists y : E, In (couple x y) r) a v0);
intros; generalize (H1 H0); clear H0 H1 H2; intros.
elim H0; clear H0; intros; auto with zfc.
Qed.
(* End of proof of lem_refl_dom_inc_img                                    *)

(***************************************************************************)
(* Si le domaine d'une relation n'est pas inclus dans l'image, alors la    *)
(* relation n'est pas reflexive.                                           *)
(***************************************************************************)
Lemma lem_refl_dom_inc_img_recip :
forall (r a : E) (rRel : rel_sig r a a),
~ inc (rdom r a a rRel) (rImg r a a rRel) -> ~ reflexivity r a rRel.