intros H3; absurd (0 < x * y)%R; auto with real.
apply Rle_not_lt;apply Rle_sign_neg_pos; auto with real.
Qed.

Theorem Rlt_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 0 y); auto with real.