
theorem HTh9:
  for V, W being non empty ModuleStr over INT.Ring, f being FrForm of V,W,
  w being Vector of W holds dom (FrFunctionalSAF(f,w)) = the
  carrier of V & rng (FrFunctionalSAF(f,w)) c= the carrier of F_Real &
  for v being Vector of V holds (FrFunctionalSAF(f,w)).v = f.(v,w)
  proof
    let V, W be non empty ModuleStr over INT.Ring, f be FrForm of V,W,
    w be Vector of W;
    set F = FrFunctionalSAF(f,w);
    dom f = [:the carrier of V,the carrier of W:] by FUNCT_2:def 1;
    then
    A1: ex g being Function st (curry' f).w = g
    & dom g = the carrier of V & rng g c= rng f &
    for y being object st y in the carrier of V holds g .y = f.(y,w)
    by FUNCT_5:32;
    hence dom F = the carrier of V & rng F c= the carrier of F_Real;
    let v be Vector of V;
    thus thesis by A1;
  end;
