(* Proof of lem_paire_not_empty                                            *)
cut (In a (paire a b)); intros.
unfold not in |- *; intros.
generalize H; clear H; rewrite H0; intros.
generalize (lem_vide_propertie a var_vide); intros; unfold Vide in H.
absurd (In a (vide var_vide)); auto with zfc.
elim (axs_paire a b a); intros; apply H0; left; reflexivity.
Qed.
(* End of proof of lem_paire_not_empty                                     *)

Lemma lem_not_vide_in_vide : ~ In Vide Vide.
unfold not in |- *; intros.