(* Proof of lem_couple_propertie                                           *)
unfold couple in |- *; intros.
unfold iff in |- *; split; intros.
elim
(lem_paire_propertie (singleton al) (paire al bl)
(singleton ar) (paire ar br)); intros.
clear H1; elim H0; clear H0; intros; [ idtac | idtac | auto with zfc ];
elim H0; clear H0; intros.
elim (lem_singleton_propertie al ar); intros.
generalize H1; clear H1; elim H2; clear H2; intros;
[ idtac | auto with zfc ].
elim (lem_paire_prop2 al bl br); intros.
split; [ reflexivity | auto with zfc ].
elim (lem_sing_eq_paire al ar br); intros.
elim (lem_sing_eq_paire ar al bl); intros.
cut (singleton ar = paire al bl); intros;
[ idtac | symmetry  in |- *; auto with zfc ].
elim H2; intros; [ idtac | auto with zfc ].
elim H4; intros; [ idtac | auto with zfc ].
rewrite <- H10; rewrite <- H7; rewrite <- H8; split; reflexivity.
elim H; clear H; intros.
rewrite H; rewrite H0; reflexivity.
Qed.
(* End of proof of lem_couple_propertie                                    *)

Lemma lem_couple_propertie_lr :
forall al ar bl br : E, couple al bl = couple ar br -> al = ar /\ bl = br.