intros P H p q f H0 H1 H2 H3 H4 H5.
generalize errorBoundedMultMin errorBoundedMultMax; intros H6 H7.
cut (MinOrMaxP b radix P);
[ intros | case H; intros H'1 (H'2, (H'3, H'4)); auto ].
case (H8 (p * q)%R f); auto.
Qed.

Theorem errorBoundedMultNeg :
forall P,
RoundedModeP b radix P ->
forall p q f : float,
Fbounded b p ->
Fbounded b q ->
(p <= 0)%R ->
(0 <= q)%R ->
(- dExp b <= Fexp p + Fexp q)%Z ->
P (p * q)%R f ->
exists r : float,
r = (p * q - f)%R :>R /\ Fbounded b r /\ Fexp r = (Fexp p + Fexp q)%Z.