(* Proof of lem_comp_dom                                                   *)
intros; apply axs_extensionnalite; intros; unfold iff in |- *; split; intros.
unfold dom in H.
elim
(axs_comprehension
(fun v0 : E =>
fun_ (comp f g a b c Af Ag) ->
exists v1 : E, In (couple v0 v1) (comp f g a b c Af Ag))
(reunion (reunion (comp f g a b c Af Ag))) v2);
intros; generalize (H0 H); clear H H0 H1; intros.
elim H; clear H; intros.
generalize (H0 (lem_comp_is_fun f g a b c Af Ag)); clear H0; intros.
elim H0; clear H0; intros.
unfold comp in H0.
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 v2 x)); intros; generalize (H1 H0);
clear H0 H1 H2; intros.
elim H0; clear H0; intros; elim H1; clear H1; intros; elim H1; clear H1;
intros.
generalize H1; clear H1 H2; rewrite (lem_first_propertie v2 x); intros.
unfold fun_sig in Af; elim Af; intros.
exact (lem_fun_and_dom f v2 x0 H2 H1).
generalize Af Ag; intros.
unfold fun_sig in Af0; unfold fun_sig in Ag0.
elim Af; clear Af; intros; elim H1; clear H1; intros.
elim Ag; clear Ag; intros; elim H4; clear H4; intros.
generalize (lem_App f v2 H0 H); intros; elim H6; clear H6; intros; elim H6;
clear H6; intros.
unfold inc in H2.
generalize (H2 x); intros.
generalize (H8 H6); clear H6 H8; intros.
cut (In x (dom g)); [ intros | rewrite H4; auto with zfc ].
generalize (lem_App g x H3 H8); intros.
elim H9; clear H9; intros; elim H9; clear H9; intros.
unfold dom in |- *.
elim
(axs_comprehension
(fun v0 : E =>
fun_ (comp f g a b c Af0 Ag0) ->
exists v1 : E, In (couple v0 v1) (comp f g a b c Af0 Ag0))
(reunion (reunion (comp f g a b c Af0 Ag0))) v2);
intros; apply H12; clear H11 H12; intros.
split.
elim (axs_reunion (reunion (comp f g a b c Af0 Ag0)) v2); intros; apply H12;
clear H11 H12; intros.
exists (singleton v2); split; [ idtac | apply lem_x_in_sing_x ].
elim (axs_reunion (comp f g a b c Af0 Ag0) (singleton v2)); intros; intros;
apply H12; clear H11 H12; intros.
exists (couple v2 x0); split;
[ idtac
| unfold couple in |- *;
elim (axs_paire (singleton v2) (paire v2 x0) (singleton v2));
intros; apply H12; left; reflexivity ].
unfold comp in |- *.
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 v2 x0)); intros; apply H12;
clear H11 H12; intros.
split; [ idtac | exists x ].
elim (lem_cartesian_propertie a c (couple v2 x0)); intros; apply H12;
clear H11 H12; intros.
exists v2; exists x0.
rewrite <- H1; split; [ auto with zfc | split; [ idtac | reflexivity ] ].
unfold inc in H5; generalize (H5 x0); intros; exact (H11 H9).
rewrite (lem_first_propertie v2 x0); rewrite (lem_second_propertie v2 x0);
split; auto with zfc.
intros; exists x0.
unfold comp in |- *.
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 v2 x0)); intros; apply H13;
clear H12 H13; intros.
split; [ idtac | exists x ].
elim (lem_cartesian_propertie a c (couple v2 x0)); intros; apply H13;
clear H12 H13; intros.
exists v2; exists x0.
rewrite <- H1; split; [ auto with zfc | split; [ idtac | reflexivity ] ].
unfold inc in H5; generalize (H5 x0); intros; exact (H12 H9).
rewrite (lem_first_propertie v2 x0); rewrite (lem_second_propertie v2 x0);
split; auto with zfc.
Qed.
(* End of proof of lem_comp_dom                                            *)

(***************************************************************************)
(* L'image de g o f est incluse dans l'image de f                          *)
(***************************************************************************)
Lemma lem_comp_img :
forall (f g a b c : E) (Af : fun_sig f a b) (Ag : fun_sig g b c),
inc (Img (comp f g a b c Af Ag)) (Img g).