(**)
intros. red. intros i j Z. apply S2. apply(Rmult_eq_reg_l c). rewrite
Rmult_comm. rewrite Z. intuition. apply Rgt_not_eq. auto. Qed.

(* Specifications: *)
Lemma simple_P12:
(forall(U:Type)(n:nat)(v:nat->R)(w:nat->Ensemble U)(S0:simple_0 _ n v w)(S1:
simple_1 _ n v w)(S2:simple_2 _ n v w)(S3:simple_3 _ n v w),
forall c:R,0<c->simple_3 _ n (fun i:nat=>v i*c) w)%R.