(* Proof of lem_rel_union_prop                                             *)
intros; unfold iff in |- *; unfold rel_union in |- *; split; intros.
elim (lem_union_propertie r1 r2 c); intros; exact (H0 H).
elim (lem_union_propertie r1 r2 c); intros; exact (H1 H).
Qed.
(* End of proof of lem_rel_union_prop                                      *)
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(* Intersection de deux relations                                          *)
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Definition rel_inter (r1 a1 b1 r2 a2 b2 : E) (rRel1 : rel_sig r1 a1 b1)
(rRel2 : rel_sig r2 a2 b2) := inter r1 r2.

(* Specifications: *)
Lemma lem_rel_inter_rel_sig :
forall (r1 a1 b1 r2 a2 b2 : E) (rRel1 : rel_sig r1 a1 b1)
(rRel2 : rel_sig r2 a2 b2),
rel_sig (rel_inter r1 a1 b1 r2 a2 b2 rRel1 rRel2)
(inter a1 a2) (inter b1 b2).