(* Proof of lem_reunion_index_prop                                         *)
intros.
generalize (lem_index_prop a I funa doma); intros.
generalize H; rewrite H0; clear H H0; intros.
elim (axs_reunion (Img a) x); intros; generalize (H0 H); clear H H0 H1;
intros.
elim H; clear H; intros; elim H; clear H; intros.
generalize (lem_App_rev a x0 funa H); intros.
elim H1; clear H1; intros; elim H1; clear H1; intros.
generalize H1; clear H1; rewrite doma; intros.
generalize (lem_eval_propertie a x1 x0 funa H2); intros.
generalize H0; clear H0; rewrite H3; clear H3; intros.
exists x1; split; auto with zfc.
Qed.
(* End of proof of lem_reunion_index_prop                                  *)

Lemma lem_index_inter :
forall (a I : E) (funa : fun_ a) (doma : dom a = I)
(Ine : I <> Vide) (x : E) (indNe : index a I funa doma <> Vide),
In x (Einter (index a I funa doma) indNe) ->
forall i : E, In i I -> In x (App a funa i).