proposition: Lemma produit_vecteur_nul : forall (a : R) (A B : PO), mult_PP a (vec A B) = zero -> a = 0 \/ A = B.
proof: Proof. unfold vec in |- *; intros. elim (classic (a = 0)); intros. left; try assumption. right; try assumption. apply conversion_PP with (a := a) (b := a); try ring; auto. cut (add_PP (cons (- a) A) (cons a B) = zero); intros. RingPP2 H1. RingPP. rewrite <- H. RingPP. Qed.