(* Proof of lem_rel_union_dom                                              *)
intros; unfold rdom in |- *; apply axs_extensionnalite; unfold iff in |- *;
split; intros.
elim
(axs_comprehension
(fun x : E =>
exists y : E, In (couple x y) (rel_union r1 a1 b1 r2 a2 b2 rRel1 rRel2))
(union a1 a2) v2); intros; generalize (H0 H); clear H H0 H1;
intros.
elim H; clear H; intros; elim H0; clear H0; intros.
unfold rel_union in H0.
elim (lem_union_propertie a1 a2 v2); intros; generalize (H1 H); clear H H1 H2;
intros.
elim (lem_union_propertie r1 r2 (couple v2 x)); intros; generalize (H1 H0);
clear H0 H1 H2; intros.
elim
(lem_union_propertie
(subset (fun x : E => exists y : E, In (couple x y) r1) a1)
(subset (fun x : E => exists y : E, In (couple x y) r2) a2) v2);
intros; apply H2; clear H1 H2; intros.
elim H; elim H0; clear H H0; intros.
left;
elim
(axs_comprehension (fun x : E => exists y : E, In (couple x y) r1) a1 v2);
intros; apply H2; clear H1 H2; split;
[ auto with zfc | exists x; auto with zfc ].
generalize (rRel2 (couple v2 x) H); intros.
elim (lem_cartesian_propertie a2 b2 (couple v2 x)); 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 v2 x0 x 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.
right;
elim
(axs_comprehension (fun x : E => exists y : E, In (couple x y) r2) a2 v2);
intros; apply H4; split; [ auto with zfc | exists x; auto with zfc ].
generalize (rRel1 (couple v2 x) H); intros.
elim (lem_cartesian_propertie a1 b1 (couple v2 x)); 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 v2 x0 x 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.
left;
elim
(axs_comprehension (fun x : E => exists y : E, In (couple x y) r1) a1 v2);
intros; apply H4; split; [ auto with zfc | exists x; auto with zfc ].
right;
elim
(axs_comprehension (fun x : E => exists y : E, In (couple x y) r2) a2 v2);
intros; apply H2; split; [ auto with zfc | exists x; auto with zfc ].
elim
(lem_union_propertie
(subset (fun x : E => exists y : E, In (couple x y) r1) a1)
(subset (fun x : E => exists y : E, In (couple x y) r2) a2) v2);
intros; generalize (H0 H); clear H H0 H1; intros.
elim
(axs_comprehension
(fun x : E =>
exists y : E, In (couple x y) (rel_union r1 a1 b1 r2 a2 b2 rRel1 rRel2))
(union a1 a2) v2); intros; apply H1; clear H0 H1;
intros.
elim H; clear H; intros.
elim
(axs_comprehension (fun x : E => exists y : E, In (couple x y) r1) a1 v2);
intros; generalize (H0 H); clear H H0 H1; intros;
elim H; clear H; intros; elim H0; clear H0; intros.
split;
[ elim (lem_union_propertie a1 a2 v2); intros; apply H2; left;
auto with zfc
| exists x; unfold rel_union in |- * ].
elim (lem_union_propertie r1 r2 (couple v2 x)); intros; apply H2; left;
auto with zfc.
elim
(axs_comprehension (fun x : E => exists y : E, In (couple x y) r2) a2 v2);
intros; generalize (H0 H); clear H H0 H1; intros;
elim H; clear H; intros; elim H0; clear H0; intros.
split;
[ elim (lem_union_propertie a1 a2 v2); intros; apply H2; right;
auto with zfc
| unfold rel_union in |- * ].
exists x; elim (lem_union_propertie r1 r2 (couple v2 x)); intros; apply H2;
right; auto with zfc.
Qed.
(* End of proof of lem_rel_union_dom                                       *)

(***************************************************************************)
(* L'image de l'union de deux relations est l'union des deux images        *)
(***************************************************************************)
Lemma lem_rel_union_Img :
forall (r1 a1 b1 r2 a2 b2 : E) (rRel1 : rel_sig r1 a1 b1)
(rRel2 : rel_sig r2 a2 b2),
rImg (rel_union r1 a1 b1 r2 a2 b2 rRel1 rRel2) (union a1 a2)
(union b1 b2) (lem_rel_union_rel_sig r1 a1 b1 r2 a2 b2 rRel1 rRel2) =
union (rImg r1 a1 b1 rRel1) (rImg r2 a2 b2 rRel2).