intros p q r H' H'0; case (Rle_or_lt 0 p); intros H'1.
repeat rewrite Rabs_right; auto with real.
apply Rle_trans with r; auto with real.
apply RmaxLess2; auto.
apply Rge_trans with p; auto with real; apply Rge_trans with q;
auto with real.
apply Rge_trans with p; auto with real.
rewrite (Rabs_left p); auto.
case (Rle_or_lt 0 q); intros H'2.
repeat rewrite Rabs_right; auto with real.
apply Rle_trans with r; auto.
apply RmaxLess2; auto.
apply Rge_trans with q; auto with real.
rewrite (Rabs_left q); auto.
case (Rle_or_lt 0 r); intros H'3.
repeat rewrite Rabs_right; auto with real.
apply Rle_trans with (- p)%R; auto with real.
apply RmaxLess1; auto.
rewrite (Rabs_left r); auto.
apply Rle_trans with (- p)%R; auto with real.
apply RmaxLess1; auto.
Qed.

Theorem Rabsolu_Zabs : forall z : Z, Rabs (IZR z) = IZR (Zabs z).
intros z; case z; simpl in |- *; auto with real.