theorem BLTh9:
  for V, W being non empty ModuleStr over INT.Ring, f being Form of V,W,
  w being Vector of W holds
  dom (FunctionalSAF(f,w)) = the carrier of V &
  rng (FunctionalSAF(f,w)) c= the carrier of INT.Ring & for v be Vector of V
  holds (FunctionalSAF(f,w)).v = f.(v,w)
  proof
    let V, W be non empty ModuleStr over INT.Ring, f be Form of V, W,
      w be Vector of W;
    set F = FunctionalSAF(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 INT.Ring;
    let v be Vector of V;
    thus thesis by A1;
  end;
