(* Proof of lem_sing_not_empty                                             *)
unfold Vide in H; generalize (lem_vide_propertie x var_vide); rewrite <- H;
intros.
generalize (lem_x_in_sing_x x); intros.
absurd (In x (singleton x)); auto with zfc.
Qed.
(* End of proof of lem_sing_not_empty                                      *)

Lemma lem_x_not_in_x : forall x : E, ~ In x x.
unfold not in |- *; intros.