intros P p H' H.
elim H'; intros H'0 H'1; elim H'1; intros H'2 H'3; elim H'3; intros H'4 H'5;
clear H'3 H'1.
case (H'0 p).
intros x H'6.
apply (H'2 p p x); auto.
apply sym_eq; apply (RoundedProjector _ _ P H'); auto.
Qed.

Theorem RoundedModeBounded :
forall P (r : R) (q : float),
RoundedModeP b radix P -> P r q -> Fbounded b q.