
theorem Th16:
  for K be non empty addLoopStr for V,W be non empty ModuleStr
  over K for f be Form of V,W, w be Vector of W holds FunctionalSAF(-f,w) = -
  FunctionalSAF(f,w)
proof
  let K be non empty addLoopStr, V,W be non empty ModuleStr over K, f be Form
  of V,W, w be Vector of W;
  now
    let v be Vector of V;
    thus (FunctionalSAF(-f,w)).v = (-f).(v,w) by Th9
      .= -f.(v,w) by Def4
      .= -(FunctionalSAF(f,w)).v by Th9
      .= (- FunctionalSAF(f,w)).v by HAHNBAN1:def 4;
  end;
  hence thesis by FUNCT_2:63;
end;
