
theorem Th9:
  for K be non empty 1-sorted for V,W be non empty ModuleStr over
  K for f be Form of V,W, w be Vector of W holds dom (FunctionalSAF(f,w)) = the
  carrier of V & rng (FunctionalSAF(f,w)) c= the carrier of K & for v be Vector
  of V holds (FunctionalSAF(f,w)).v = f.(v,w)
proof
  let K be non empty 1-sorted, V,W be non empty ModuleStr over K, 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 be Function st (curry' f).w = g & dom g = the carrier of V & rng g
  c= rng f & for y be 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 K;
  let v be Vector of V;
  thus thesis by A1;
end;
