
theorem HTh8:
  for V, W being non empty ModuleStr over INT.Ring, f being FrForm of V,W,
  v being Vector of V
  holds dom (FrFunctionalFAF(f,v)) = the
  carrier of W & rng (FrFunctionalFAF(f,v)) c= the carrier of F_Real
  & for w being Vector of W holds (FrFunctionalFAF(f,v)).w = f.(v,w)
  proof
    let V, W be non empty ModuleStr over INT.Ring;
    let f be FrForm of V,W, v be Vector of V;
    set F = FrFunctionalFAF(f,v);
    dom f = [:the carrier of V,the carrier of W:] by FUNCT_2:def 1;
    then
    A1: ex g being Function st (curry f).v = g
    & dom g = the carrier of W & rng g c= rng f &
    for y being object st y in the carrier of W holds g.y = f.( v,y)
    by FUNCT_5:29;
    hence dom F = the carrier of W & rng F c= the carrier of F_Real;
    let y be Vector of W;
    thus thesis by A1;
  end;
