intros b p q H' H'0 H'1; case (Zle_or_lt (Fexp p) (Fexp q)); intros Rl0.
exists p; auto.
cut ((Fexp p - Zabs_nat (Fexp p - Fexp q))%Z = Fexp q);
[ intros Eq1 | idtac ].
exists (Fshift radix (Zabs_nat (Fexp p - Fexp q)) p); split; split; auto.
apply Zle_lt_trans with (Fnum q); auto with float.
replace (Zabs (Fnum (Fshift radix (Zabs_nat (Fexp p - Fexp q)) p))) with
(Fnum (Fabs (Fshift radix (Zabs_nat (Fexp p - Fexp q)) p)));
auto.
apply (Rle_Fexp_eq_Zle radix); auto with arith.
rewrite Fabs_correct; auto with arith; rewrite FshiftCorrect; auto with arith;
rewrite <- (Fabs_correct radix); auto with float arith.
rewrite <- (Zabs_eq (Fnum q)); auto with float zarith.
apply (LeR0Fnum radix); auto.
apply Rle_trans with (2 := H'1); auto with real.
rewrite (Fabs_correct radix); auto with real zarith.
unfold Fshift in |- *; simpl in |- *; rewrite Eq1; auto with float.
unfold FtoRradix in |- *; apply FshiftCorrect; auto.
unfold Fshift in |- *; simpl in |- *.
rewrite Eq1; auto with zarith.
rewrite inj_abs; auto with zarith; ring.
Qed.

Theorem Zle_monotony_contra_abs_pow :
forall x y z n : Z,
(0 < z)%Z ->
(Rabs (x * powerRZ z n) <= Rabs (y * powerRZ z n))%R -> (Zabs x <= Zabs y)%Z.