proposition: Lemma vecteur_homothetie : forall (k : R) (I A A' : PO), vec I A' = mult_PP k (vec I A) :>PP -> A' = homothetie k I A :>PO.
proof: Proof. unfold vec, homothetie in |- *; intros. elim cons_inj with (a := 1) (b := k + (1 + - k)) (A := A') (B := barycentre (cons k A) (cons (1 + - k) I)); intros; auto with *. repeat rewrite <- add_PP_barycentre; try discrR. RingPP2 H. RingPP. replace (k + (1 + - k)) with 1; try ring; try discrR. Qed.