(* Proof of lem_union_propertie                                            *)
intros; unfold union in |- *.
elim (axs_reunion (paire a b) v0); intros.
unfold iff in |- *; split; intros.
elim H; clear H; intros.
clear H0; clear H1.
elim (axs_paire a b x); unfold iff in |- *; intros.
elim H; clear H; intros.
clear H1; elim H0; clear H0; intros.
clear H; rewrite <- H0; left; assumption.
clear H; rewrite <- H0; right; assumption.
clear H2; assumption.
clear H0; assumption.
apply H0; elim H1; clear H1; intros.
clear H; clear H0; exists a; split.
clear H1; elim (axs_paire a b a); intros; auto with zfc.
assumption.
clear H; clear H0; exists b; split.
clear H1; elim (axs_paire a b b); intros; auto with zfc.
assumption.
Qed.
(* End of proof of lem_union_propertie                                     *)

Lemma lem_union_propertie_lr :
forall a b v0 : E, In v0 (union a b) -> In v0 a \/ In v0 b.