(* Proof of lem_fun_equiv_is_rel                                           *)
unfold rel_sig in |- *; unfold inc in |- *; unfold equiv_fun in |- *; intros.
elim
(axs_comprehension
(fun x : E =>
exists a : E, (exists b : E, x = couple a b /\ App f Af a = App f Af b))
(cartesien (dom f) (dom f)) v0); intros; generalize (H0 H);
clear H H0 H1; intros.
elim H; clear H; intros; auto with zfc.
Qed.
(* End of proof of lem_fun_equiv_is_rel                                    *)

Lemma lem_sig_fun_equiv_is_rel :
forall (f a s : E) (Afs : fun_sig f a s),
rel_sig (equiv_fun f (lem_fun_sig_is_fun f a s Afs)) a a.