
theorem
  for V, W being non empty ModuleStr over INT.Ring, f, g being FrForm of V,W,
  w being Vector of W holds
  FrFunctionalSAF(f-g,w) = FrFunctionalSAF(f,w) - FrFunctionalSAF(g,w)
  proof
    let V, W be non empty ModuleStr over INT.Ring, f, g be FrForm of V,W,
    w be Vector of W;
    now
      let v be Vector of V;
      thus (FrFunctionalSAF(f-g,w)).v = (f-g).(v,w) by HTh9
      .= f.(v,w) - g.(v,w) by Def7
      .= (FrFunctionalSAF(f,w)).v - g.(v,w) by HTh9
      .= (FrFunctionalSAF(f,w)).v - (FrFunctionalSAF(g,w)).v by HTh9
      .= (FrFunctionalSAF(f,w)).v + (-FrFunctionalSAF(g,w)).v by HDef4
      .= (FrFunctionalSAF(f,w) -FrFunctionalSAF(g,w)).v by HDef3;
    end;
    hence thesis by FUNCT_2:63;
  end;
