intros p q r H; unfold Rmax in |- *.
case (Rle_dec p q); case (Rle_dec (r * p) (r * q)); auto; intros H1 H2; auto.
case H; intros E1.
case H1; auto with real.
rewrite <- E1; repeat rewrite Rmult_0_l; auto.
case H; intros E1.
case H2; auto with real.
apply Rmult_le_reg_l with (r := r); auto.
rewrite <- E1; repeat rewrite Rmult_0_l; auto.
Qed.

Theorem Rle_R0_Ropp : forall p : R, (p <= 0)%R -> (0 <= - p)%R.
intros p H; rewrite <- Ropp_0; auto with real.