intros P H' p q r H'0 H'1 H'2; case H'1; intros H'3.
cut (MonotoneP radix P);
[ intros Mn | apply RoundedModeP_inv4 with (1 := H'); auto ].
apply (Mn r p); auto.
apply RoundedModeProjectorIdem with (P := P); auto.
rewrite RoundedModeProjectorIdemEq with (P := P) (p := p) (q := q);
auto with real.
cut (CompatibleP b radix P);
[ intros Cp | apply RoundedModeP_inv2 with (1 := H'); auto ].
apply (Cp r p q); auto.
apply RoundedModeBounded with (P := P) (r := r); auto.
Qed.

Theorem RoundAbsMonotoner :
forall (P : R -> float -> Prop) (p : R) (q r : float),
RoundedModeP b radix P ->
Fbounded b r -> P p q -> (Rabs p <= r)%R -> (Rabs q <= r)%R.