Definition double_round_mult_hyp fexp1 fexp2 :=
  (forall ex ey, (fexp2 (ex + ey) <= fexp1 ex + fexp1 ey)%Z)
  /\ (forall ex ey, (fexp2 (ex + ey - 1) <= fexp1 ex + fexp1 ey)%Z).

Theorem double_round_mult_FTZ : (emin' + prec' <= 2 * emin + prec)%Z -> (2 * prec <= prec')%Z -> forall x y, FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> round beta (FTZ_exp emin prec) rnd (round beta (FTZ_exp emin' prec') rnd (x * y)) = round beta (FTZ_exp emin prec) rnd (x * y).