theorem
  for V, W being non empty ModuleStr over INT.Ring, f, g being Form of V,W,
  w being Vector of W
  holds FunctionalSAF(f-g,w) = FunctionalSAF(f,w) - FunctionalSAF(g,w)
  proof
    let V, W be non empty ModuleStr over INT.Ring, f,g be Form of V,W,
      w be Vector of W;
    now
      let v be Vector of V;
      thus (FunctionalSAF(f-g,w)).v = (f-g).(v,w) by BLTh9
      .= f.(v,w) - g.(v,w) by BLDef7
      .= (FunctionalSAF(f,w)).v - g.(v,w) by BLTh9
      .= (FunctionalSAF(f,w)).v - (FunctionalSAF(g,w)).v by BLTh9
      .= (FunctionalSAF(f,w)).v + (-FunctionalSAF(g,w)).v by HAHNBAN1:def 4
      .= (FunctionalSAF(f,w) -FunctionalSAF(g,w)).v by HAHNBAN1:def 3;
    end;
    hence thesis by FUNCT_2:63;
  end;
