(* Proof of lem_rId_dom                                                    *)
elim
(axs_comprehension (fun x : E => exists y : E, In (couple x y) (rId a)) a v2);
intros; generalize (H0 H); clear H H0 H1; intros;
elim H; clear H; intros; auto with zfc.
elim
(axs_comprehension (fun x : E => exists y : E, In (couple x y) (rId a)) a v2);
intros; apply H1; clear H0 H1; split;
[ auto with zfc | exists v2; unfold rId in |- * ].
elim
(axs_comprehension (fun c : E => exists x : E, c = couple x x)
(cartesien a a) (couple v2 v2)); intros; apply H1;
clear H0 H1; intros.
split;
[ elim (lem_cartesian_propertie a a (couple v2 v2)); intros; apply H1;
exists v2; exists v2; split;
[ auto with zfc | split; [ auto with zfc | reflexivity ] ]
| exists v2; reflexivity ].
Qed.
(* End of proof of lem_rId_dom                                             *)
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(* L'image de l'identite est a                                             *)
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(* Lemma lem_rId_Img : (a:E)(rImg (rId a) a a (lem_rId_rel_sig a))=a . *)
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(* Complementaire d'une relation                                           *)
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Definition rel_Comp (r a b : E) (rRel : rel_sig r a b) :=
Comp (cartesien a b) r.

(* Specifications: *)
Lemma lem_rel_Comp_rel_sig :
forall (r a b : E) (rRel : rel_sig r a b), rel_sig (rel_Comp r a b rRel) a b.