(* Proof of lem_inter_propertie                                            *)
intros; unfold iff in |- *; unfold inter in |- *; unfold def_F_inter in |- *;
split; intros.
elim (axs_comprehension (def_F_inter b) a x); unfold def_F_inter in |- *;
intros; auto with zfc.
elim (axs_comprehension (def_F_inter b) a x); unfold def_F_inter in |- *;
intros; auto with zfc.
Qed.
(* End of proof of lem_inter_propertie                                     *)

Lemma lem_inter_propertie_lr :
forall a b x : E, In x (inter a b) -> In x a /\ In x b.