
theorem
  for K be non empty addLoopStr for V,W be non empty ModuleStr over K
  for f,g be Form of V,W, v be Vector of V holds FunctionalFAF(f-g,v) =
  FunctionalFAF(f,v) - FunctionalFAF(g,v)
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 V;
  now
    let v be Vector of W;
    thus (FunctionalFAF(f-g,w)).v = (f-g).(w,v) by Th8
      .= f.(w,v) - g.(w,v) by Def7
      .= (FunctionalFAF(f,w)).v - g.(w,v) by Th8
      .= (FunctionalFAF(f,w)).v - (FunctionalFAF(g,w)).v by Th8
      .= (FunctionalFAF(f,w)).v +- (FunctionalFAF(g,w)).v by RLVECT_1:def 11
      .= (FunctionalFAF(f,w)).v + (-FunctionalFAF(g,w)).v by HAHNBAN1:def 4
      .= (FunctionalFAF(f,w) -FunctionalFAF(g,w)).v by HAHNBAN1:def 3;
  end;
  hence thesis by FUNCT_2:63;
end;
