intros P H' r q q' H'0 H'1.
replace (radix * q')%R with (FtoRradix (Float (Fnum q') (Fexp q' + 1%nat))).
intros H'2; case H'2.
intros H'3; case H'; intros H'4 H'5; elim H'5; intros H'6 H'7; elim H'7;
intros H'8 H'9; apply H'9 with (1 := H'3); clear H'7 H'5;
auto.
apply RoundedModeProjectorIdem; auto.
apply FBoundedScale; auto.
intros H'3.
replace (FtoRradix q) with (FtoRradix (Float (Fnum q') (Fexp q' + 1%nat)));
auto with real.
apply (RoundedProjector _ _ P H'); auto.
apply FBoundedScale; auto.
case H'.
intros H'4 H'5; elim H'5; intros H'6 H'7; clear H'5.
apply (H'6 r (Float (Fnum q') (Fexp q' + 1%nat)) q); auto.
apply RoundedModeBounded with (P := P) (r := r); auto.
rewrite (FvalScale _ radixMoreThanOne b).
rewrite powerRZ_1; auto.
Qed.

Theorem RoundedModeMultLess :
forall P,
RoundedModeP b radix P ->
forall (r : R) (q q' : float),
P r q -> Fbounded b q' -> (radix * q' <= r)%R -> (radix * q' <= q)%R.