(* Proof of lem_x_in_equivClass_x                                          *)
intros; unfold equivClass in |- *.
elim (axs_comprehension (fun y : E => In (couple x y) r) a x); intros;
apply H1; clear H0 H1; split; [ auto with zfc | idtac ].
elim rEqv; clear rEqv; intros.
unfold reflexivity in H0.
generalize H; rewrite <- rD; clear H; intros.
exact (H0 x H).
Qed.
(* End of proof of lem_x_in_equivClass_x                                   *)

(* Specifications: *)
(***************************************************************************)
(* Le noyau d'equivalence de projQuotient est r                            *)
(***************************************************************************)
Lemma lem_projQuotient_equiv_fun :
forall (r a : E) (rRel : rel_sig r a a) (rEqv : equivalence r a rRel)
(rD : rdom r a a rRel = a),
equiv_fun (projQuotient r a rRel rEqv rD)
(lem_projQuotient_is_fun r a rRel rEqv rD) = r.