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 p r); 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 RleBoundRoundr :
forall P,
RoundedModeP b radix P ->
forall (p q : float) (r : R),
Fbounded b p -> (r <= p)%R -> P r q -> (q <= p)%R.