intros n m p H1 H2; replace (p * n)%R with (-(-p * n))%R; auto with real; try ring.
replace (p * m)%R with (-(-p * m))%R; auto with real; try ring.
Qed.

(* Specifications: *)
Theorem Ropp_lt: forall n m, (m < n -> -n < -m)%R.
auto with real.
Theorem Ropp_ge: forall n m, (m >= n -> -n >= -m)%R.
auto with real.
Theorem Ropp_lt: forall n m, (m < n -> -n < -m)%R.
auto with real.
Theorem Ropp_ge: forall n m, (m >= n -> -n >= -m)%R.
auto with real.