theorem
  for V, W being non empty ModuleStr over INT.Ring, f being Form of V,W,
  w being Vector of W
  holds FunctionalSAF(-f,w) = -FunctionalSAF(f,w)
  proof
    let V, W be non empty ModuleStr over INT.Ring, 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 BLTh9
      .= -f.(v,w) by BLDef4
      .= -(FunctionalSAF(f,w)).v by BLTh9
      .= (- FunctionalSAF(f,w)).v by HAHNBAN1:def 4;
    end;
    hence thesis by FUNCT_2:63;
  end;
