(* Proof of lem_inter_commut                                               *)
elim (lem_inter_propertie a b v2); intros.
elim (lem_inter_propertie b a v2); intros.
unfold iff in |- *; split; intros.
clear H0; clear H1; apply H2; apply lem_and_sym; apply H; auto with zfc.
clear H; clear H2; apply H0; apply lem_and_sym; apply H1; auto with zfc.
Qed.
(* End of proof of lem_inter_commut                                        *)

Lemma lem_inter_assoc :
forall a b c : E, inter (inter a b) c = inter a (inter b c).