
theorem
  for V, W being non empty ModuleStr over INT.Ring, f being FrForm of V,W,
  w being Vector of W holds
  FrFunctionalSAF(-f,w) = -FrFunctionalSAF(f,w)
  proof
    let V, W be non empty ModuleStr over INT.Ring, f be FrForm of V,W,
    w be Vector of W;
    now
      let v be Vector of V;
      thus (FrFunctionalSAF(-f,w)).v = (-f).(v,w) by HTh9
      .= -f.(v,w) by Def4
      .= -(FrFunctionalSAF(f,w)).v by HTh9
      .= (- FrFunctionalSAF(f,w)).v by HDef4;
    end;
    hence thesis by FUNCT_2:63;
  end;
