(* Proof of lem_comp_is_fun                                                *)
intros.
unfold fun_ in |- *; split; [ exists a; exists c | intros ].
unfold inc in |- *; intros.
unfold comp in H.
elim
(axs_comprehension
(fun c0 : E =>
exists y : E, In (couple (first c0) y) f /\ In (couple y (second c0)) g)
(cartesien a c) v0); intros; generalize (H0 H);
clear H H0 H1; intros.
elim H; clear H; intros; elim H0; clear H0; intros; elim H0; clear H0; intros.
unfold fun_sig in Af; unfold fun_sig in Ag.
elim Af; clear Af; intros.
elim H3; clear H3; intros.
elim Ag; clear Ag; intros.
elim H6; clear H6; intros.
auto with zfc.
unfold comp in H; unfold comp in H0.
elim
(axs_comprehension
(fun c0 : E =>
exists y0 : E,
In (couple (first c0) y0) f /\ In (couple y0 (second c0)) g)
(cartesien a c) (couple x y)); intros; generalize (H1 H);
clear H H1 H2; intros.
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 z)); intros; generalize (H1 H0);
clear H0 H1 H2; intros.
elim H; clear H; intros; elim H1; clear H1; intros; elim H1; clear H1; intros.
elim H0; clear H0; intros; elim H3; clear H3; intros; elim H3; clear H3;
intros.
generalize H3; clear H3; rewrite (lem_first_propertie x z); intros.
generalize H4; clear H4; rewrite (lem_second_propertie x z); intros.
generalize H1; clear H1; rewrite (lem_first_propertie x y); intros.
generalize H2; clear H2; rewrite (lem_second_propertie x y); intros.
unfold fun_sig in Af; elim Af; intros.
unfold fun_sig in Ag; elim Ag; intros.
unfold fun_ in H5; unfold fun_ in H7; clear H6 H8.
elim H5; clear H5; intros; clear H5.
elim H7; clear H7; intros; clear H5.
generalize (H6 x x0 x1 H1 H3); clear H1 H3; intros.
generalize H4; clear H4; rewrite <- H1; clear H1; intros.
exact (H7 x0 y z H2 H4).
Qed.
(* End of proof of lem_comp_is_fun                                         *)

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(* Le domaine de g o f est le domaine de f                                 *)
(***************************************************************************)
Lemma lem_comp_dom :
forall (f g a b c : E) (Af : fun_sig f a b) (Ag : fun_sig g b c),
dom (comp f g a b c Af Ag) = dom f.