case (Rle_dec p q); case (Rle_dec q p); auto; intros H1 H2; apply Rle_antisym;
auto.
case (Rle_or_lt p q); auto; intros H'0; Contradict H1; apply Rlt_le; auto.
case (Rle_or_lt q p); auto; intros H'0; Contradict H2; apply Rlt_le; auto.
Qed.

Theorem RmaxAbs :
forall p q r : R,
(p <= q)%R -> (q <= r)%R -> (Rabs q <= Rmax (Rabs p) (Rabs r))%R.