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

(***************************************************************************)
(* Des evidences                                                           *)
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Lemma lem_rel_union_prop :
forall (r1 a1 b1 r2 a2 b2 : E) (rRel1 : rel_sig r1 a1 b1)
(rRel2 : rel_sig r2 a2 b2) (c : E),
In c (rel_union r1 a1 b1 r2 a2 b2 rRel1 rRel2) <-> In c r1 \/ In c r2.