(* Proof of lem_union_vide                                                 *)
elim (lem_union_propertie a Vide v2); intros.
unfold iff in |- *; split; intros.
elim H; clear H H0; intros;
[ auto with zfc
| unfold Vide in H1; elim (lem_vide_propertie v2 var_vide);
auto with zfc
| auto with zfc ].
apply H0; clear H H0; left; auto with zfc.
Qed.
(* End of proof of lem_union_vide                                          *)

Lemma lem_inter_vide : forall a : E, inter a Vide = Vide.
intros; apply axs_extensionnalite; intros.