intros r1 r2 min max H' H'0 H'1 H'2; split.
case H'0; auto.
split.
apply Rlt_le; auto.
intros f H'3 H'4.
case H'0; auto.
intros H'5 H'6; elim H'6; intros H'7 H'8; apply H'8; clear H'6; auto.
case (Rle_or_lt r1 (FtoR radix f)); auto; intros C1.
absurd (min < FtoR radix f)%R.
apply Rle_not_lt.
case H'.
intros H'6 H'9; elim H'9; intros H'10 H'11; apply H'11; clear H'9; auto.
apply Rlt_le; auto.
apply Rlt_le_trans with (1 := H'1); auto.
Qed.

(* Specifications: *)
Theorem roundedModeLessMult :
forall (P : R -> float -> Prop) (p : float) (r : R),
RoundedModeP b radix P ->
P r p -> (Float 1%nat (- dExp b) <= r)%R -> (p <= radix * r)%R.