(* Proof of lem_rel_inv_dom                                                *)
unfold rdom in |- *; unfold rImg in |- *; intros; apply axs_extensionnalite;
unfold iff in |- *; split; intros.
elim
(axs_comprehension
(fun x : E => exists y : E, In (couple x y) (rel_inverse r a b rRel)) b
v2); intros; generalize (H0 H); clear H H0 H1;
intros; elim H; clear H; intros.
elim H0; clear H0; unfold rel_inverse in |- *; intros.
elim
(axs_comprehension
(fun c : E =>
exists x0 : E, (exists y : E, In (couple x0 y) r /\ c = couple y x0))
(cartesien b a) (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; elim H1; clear H1; intros.
elim (lem_couple_propertie v2 x1 x x0); intros; generalize (H3 H2);
clear H2 H3 H4; intros; elim H2; clear H2; intros.
generalize H1; rewrite <- H2; rewrite <- H3; clear H1 H2 H3; clear x0 x1;
intros.
elim (axs_comprehension (fun y : E => exists x : E, In (couple x y) r) b v2);
intros; apply H3; clear H2 H3; intros.
split; [ auto with zfc | exists x; auto with zfc ].
elim (axs_comprehension (fun y : E => exists x : E, In (couple x y) r) b v2);
intros; generalize (H0 H); clear H H0 H1; intros;
elim H; clear H; intros; elim H0; clear H0; intros.
elim
(axs_comprehension
(fun x : E => exists y : E, In (couple x y) (rel_inverse r a b rRel)) b
v2); intros; apply H2; clear H1 H2; intros.
split; [ auto with zfc | unfold rel_inverse in |- *; exists x ].
elim
(axs_comprehension
(fun c : E =>
exists x0 : E, (exists y : E, In (couple x0 y) r /\ c = couple y x0))
(cartesien b a) (couple v2 x)); intros; apply H2;
clear H1 H2; intros.
split;
[ idtac | exists x; exists v2; split; [ auto with zfc | reflexivity ] ].
generalize (rRel (couple x v2) H0); intros.
elim (lem_cartesian_propertie a b (couple x v2)); intros; generalize (H2 H1);
clear H1 H2 H3; intros.
elim H1; clear H1; intros; elim H1; clear H1; intros; elim H1; clear H1;
intros; elim H2; clear H2; intros.
elim (lem_couple_propertie x x0 v2 x1); intros; generalize (H4 H3);
clear H3 H4 H5; intros; elim H3; clear H3; intros;
generalize H1 H2; rewrite <- H3; rewrite <- H4; clear H1 H2 H3 H4;
clear x0 x1; intros.
elim (lem_cartesian_propertie b a (couple v2 x)); intros; apply H4;
clear H3 H4; intros.
exists v2; exists x; split;
[ auto with zfc | split; [ auto with zfc | reflexivity ] ].
Qed.
(* End of proof of lem_rel_inv_dom                                         *)

(***************************************************************************)
(* L'image de la relation inverse est le domaine de la relation initiale   *)
(***************************************************************************)
Lemma lem_rel_inv_Img :
forall (r a b : E) (rRel : rel_sig r a b),
rImg (rel_inverse r a b rRel) b a (lem_rel_inv_is_rel_sig r a b rRel) =
rdom r a b rRel.