(* Proof of lem_rel_inter_Img                                              *)
unfold inc in |- *; intros.
unfold rImg in H.
elim
(axs_comprehension
(fun y : E =>
exists x : E, In (couple x y) (rel_inter r1 a1 b1 r2 a2 b2 rRel1 rRel2))
(inter b1 b2) v0); intros; generalize (H0 H); clear H H0 H1;
intros; elim H; clear H; intros; elim H0; clear H0;
intros.
elim (lem_inter_propertie r1 r2 (couple x v0)); intros; generalize (H1 H0);
clear H0 H1 H2; intros; elim H0; clear H0; intros.
generalize (lem_rel_and_rImg r1 a1 b1 rRel1 x v0 H0)
(lem_rel_and_rImg r2 a2 b2 rRel2 x v0 H1); clear H0 H1;
intros;
elim (lem_inter_propertie (rImg r1 a1 b1 rRel1) (rImg r2 a2 b2 rRel2) v0);
intros; apply H3; split; auto with zfc.
Qed.
(* End of proof of lem_rel_inter_Img                                       *)

(* Specifications: *)
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(* Une evidence                                                            *)
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Lemma lem_rel_inter_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_inter r1 a1 b1 r2 a2 b2 rRel1 rRel2) <-> In c r1 /\ In c r2.