
theorem
  for K be non empty addLoopStr for V,W be non empty ModuleStr over K
  for f,g be Form of V,W, w be Vector of W holds FunctionalSAF(f-g,w) =
  FunctionalSAF(f,w) - FunctionalSAF(g,w)
proof
  let K be non empty addLoopStr, V,W be non empty ModuleStr over K, f,g be
  Form of V,W, w be Vector of W;
  now
    let v be Vector of V;
    thus (FunctionalSAF(f-g,w)).v = (f-g).(v,w) by Th9
      .= f.(v,w) - g.(v,w) by Def7
      .= (FunctionalSAF(f,w)).v - g.(v,w) by Th9
      .= (FunctionalSAF(f,w)).v - (FunctionalSAF(g,w)).v by Th9
      .= (FunctionalSAF(f,w)).v +- (FunctionalSAF(g,w)).v by RLVECT_1:def 11
      .= (FunctionalSAF(f,w)).v + (-FunctionalSAF(g,w)).v by HAHNBAN1:def 4
      .= (FunctionalSAF(f,w) -FunctionalSAF(g,w)).v by HAHNBAN1:def 3;
  end;
  hence thesis by FUNCT_2:63;
end;
