(* Proof of lem_comp_img                                                   *)
intros; unfold inc in |- *; intros.
elim Af; intros; elim H1; clear H1; intros.
elim Ag; intros; elim H4; clear H4; intros.
elim
(lem_App_rev (comp f g a b c Af Ag) v0 (lem_comp_is_fun f g a b c Af Ag) H);
intros.
elim H6; clear H6; intros.
unfold comp in H7.
elim
(axs_comprehension
(fun c0 : E =>
exists y : E, In (couple (first c0) y) f /\ In (couple y (second c0)) g)
(cartesien a c) (couple x v0)); intros; generalize (H8 H7);
clear H7 H8 H9; intros.
elim H7; clear H7; intros; elim H8; clear H8; intros; elim H8; clear H8;
intros.
generalize H9; clear H9; rewrite (lem_second_propertie x v0); intros.
exact (lem_fun_and_img g x0 v0 H3 H9).
Qed.
(* End of proof of lem_comp_img                                            *)

(***************************************************************************)
(* (x,y) in f et (y,z) in g -> (x,z) in g o f                              *)
(***************************************************************************)
Lemma lem_comp_make :
forall (f g a b c x y z : E) (Af : fun_sig f a b) (Ag : fun_sig g b c),
In (couple x y) f ->
In (couple y z) g -> In (couple x z) (comp f g a b c Af Ag).