
theorem Th17:
  for K be non empty addLoopStr for V,W be non empty ModuleStr
  over K for f be Form of V,W, v be Vector of V holds FunctionalFAF(-f,v) = -
  FunctionalFAF(f,v)
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 V;
  now
    let v be Vector of W;
    thus (FunctionalFAF(-f,w)).v = (-f).(w,v) by Th8
      .= -f.(w,v) by Def4
      .= -(FunctionalFAF(f,w)).v by Th8
      .= (- FunctionalFAF(f,w)).v by HAHNBAN1:def 4;
  end;
  hence thesis by FUNCT_2:63;
end;
