intros H3; absurd (0 <= x * y)%R; auto with real.
apply Rlt_not_le;apply Rlt_sign_neg_pos; auto.
Qed.

(* Specifications: *)
Theorem Rle_sign_pos_neg_rev: forall x y: R, (0 < x -> x * y <= 0 -> y <= 0)%R.
intros x y H1 H2; case (Rle_or_lt y 0); auto with real.
Theorem Rle_sign_pos_neg_rev: forall x y: R, (0 < x -> x * y <= 0 -> y <= 0)%R.
intros x y H1 H2; case (Rle_or_lt y 0); auto with real.