
theorem
  for V, W being non empty ModuleStr over INT.Ring, f being FrForm of V,W,
  v being Vector of V holds
  FrFunctionalFAF(-f,v) = -FrFunctionalFAF(f,v)
  proof
    let V, W be non empty ModuleStr over INT.Ring, f be FrForm of V,W,
    w be Vector of V;
    now
      let v be Vector of W;
      thus (FrFunctionalFAF(-f,w)).v = (-f).(w,v) by HTh8
      .= -f.(w,v) by Def4
      .= -(FrFunctionalFAF(f,w)).v by HTh8
      .= (- FrFunctionalFAF(f,w)).v by HDef4;
    end;
    hence thesis by FUNCT_2:63;
  end;
