proposition: Lemma rec_translation_vecteur : forall I J A A' : PO, vec I J = vec A A' :>PP -> A' = translation I J A :>PO.
proof: Proof. unfold vec, translation in |- *; intros. elim cons_inj with (a := 1) (b := -1 + 2) (A := A') (B := barycentre (cons (-1) I) (cons 2 (barycentre (cons 1 J) (cons 1 A)))); intros; auto with *. replace 2 with (1 + 1) at 2 by ring. repeat rewrite <- add_PP_barycentre; try discrR. RingPP1 H. RingPP. Qed.