proposition: Lemma add_PP_vecteur : forall (a b : R) (A B C M : PO), a + b <> 0 -> add_PP (cons a A) (cons b B) = cons (a + b) C -> add_PP (mult_PP a (vec M A)) (mult_PP b (vec M B)) = mult_PP (a + b) (vec M C).
proof: Proof. unfold vec in |- *; intros. repeat rewrite <- distrib_mult_cons; auto. replace (a * 1) with a; try ring; auto. RingPP1 H0. RingPP. Qed.