theorem
  for V, W being non empty ModuleStr over INT.Ring, f being Form of V,W,
  v being Vector of V
  holds FunctionalFAF(-f,v) = -FunctionalFAF(f,v)
  proof
    let V, W be non empty ModuleStr over INT.Ring, 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 BLTh8
      .= -f.(w,v) by BLDef4
      .= -(FunctionalFAF(f,w)).v by BLTh8
      .= (- FunctionalFAF(f,w)).v by HAHNBAN1:def 4;
    end;
    hence thesis by FUNCT_2:63;
  end;
