
theorem Th8:
  for K be non empty 1-sorted for V,W be non empty ModuleStr over K
  for f be Form of V,W, v be Vector of V holds dom (FunctionalFAF(f,v)) = the
  carrier of W & rng (FunctionalFAF(f,v)) c= the carrier of K & for w be Vector
  of W holds (FunctionalFAF(f,v)).w = f.(v,w)
proof
  let K be non empty 1-sorted, V,W be non empty ModuleStr over K;
  let f be Form of V,W, v be Vector of V;
  set F = FunctionalFAF(f,v);
  dom f = [:the carrier of V,the carrier of W:] by FUNCT_2:def 1;
  then
A1: ex g be Function st (curry f).v = g & dom g = the carrier of W & rng g c=
  rng f & for y be 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 K;
  let y be Vector of W;
  thus thesis by A1;
end;
