intros P p q H' H'0 H'1.
cut (MinOrMaxP b radix P);
[ intros Mn; case (Mn p q); auto; intros Mn1 | auto with inv ].
apply sym_eq; apply MinEq with (1 := Mn1); auto.
apply (RoundedModeProjectorIdem (isMin b radix)); auto.
apply MinRoundedModeP with (precision := precision); auto.
apply sym_eq; apply MaxEq with (1 := Mn1); auto.
apply (RoundedModeProjectorIdem (isMax b radix)); auto.
apply MaxRoundedModeP with (precision := precision); auto.
Qed.

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