(* Proof of lem_eval_prop2                                                 *)
unfold dom in |- *; intros.
elim
(axs_comprehension
(fun v0 : E => fun_ f -> exists v1 : E, In (couple v0 v1) f)
(reunion (reunion f)) x); intros; generalize (H0 H);
clear H H0 H1; intros.
elim H; clear H; intros.
generalize (H0 Af); clear H0; intros; elim H0; clear H0; intros.
generalize (lem_eval_propertie f x x0 Af H0); intros.
rewrite <- H1; auto with zfc.
Qed.
(* End of proof of lem_eval_prop2                                          *)

(* Specifications: *)
(***************************************************************************)
(* Si v2 est dans f, alors v2 est un couple ...                            *)
(***************************************************************************)
Lemma lem_x_in_fun_form :
forall (f : E) (Af : fun_ f) (x : E),
In x f -> exists a : E, (exists b : E, x = couple a b).