Definition couple (a b : E) := paire (singleton a) (paire a b).

(* Specifications: *)
Lemma lem_paire_prop2 : forall a b c : E, paire a b = paire a c <-> b = c.
Lemma lem_sing_eq_paire : forall a b c : E, singleton a = paire b c <-> a = b /\ a = c.
Lemma lem_couple_propertie : forall al ar bl br : E, couple al bl = couple ar br <-> al = ar /\ bl = br.
Lemma lem_couple_propertie_lr : forall al ar bl br : E, couple al bl = couple ar br -> al = ar /\ bl = br.
Lemma lem_couple_propertie_rl : forall al ar bl br : E, al = ar /\ bl = br -> couple al bl = couple ar br.