intros b f n m H' H'0; split; auto.
simpl in |- *; auto.
apply Zle_lt_trans with (Zabs (Fnum (Fshift radix n f))).
simpl in |- *; replace m with (m + 0); auto with arith.
replace n with (m + (n - m)); auto with arith.
repeat rewrite Zpower_nat_is_exp.
repeat rewrite Zabs_Zmult; auto.
apply Zle_Zmult_comp_l; auto with zarith.
apply Zle_Zmult_comp_l; auto with zarith.
repeat rewrite Zabs_eq; auto with zarith.
case H'0; auto.
apply Zle_trans with (Fexp (Fshift radix n f)); auto with float.
simpl in |- *; unfold Zminus in |- *; auto with zarith.
Qed.

(* Specifications: *)
Theorem eqExpMax :
forall (b : Fbound) (p q : float),
Fbounded b p ->
Fbounded b q ->
(Fabs p <= q)%R ->
exists r : float, Fbounded b r /\ r = p :>R /\ (Fexp r <= Fexp q)%Z.