apply Rle_lt_trans with (1 := H').
unfold boundR, boundNat, FtoRradix, FtoR in |- *; simpl in |- *;
auto with real.
rewrite Rmult_1_l; auto with real zarith.
apply Rlt_trans with (2 := boundNatCorrect (Zabs_nat (up (Rabs r)))).
replace (Rabs r) with r; auto with real.
apply Rlt_le_trans with (r2 := IZR (up r)); auto with real zarith.
case (archimed r); auto.
rewrite INR_IZR_INZ; auto with real zarith.
unfold Rabs in |- *; case (Rcase_abs r); auto with real.
intros H'0; Contradict H'0; auto with real.
Qed.

Theorem boundRrOpp : forall r : R, boundR r = boundR (- r).
intros R; unfold boundR in |- *.