
theorem
  for V, W being non empty ModuleStr over INT.Ring, f, g being FrForm of V,W,
  v being Vector of V holds
  FrFunctionalFAF(f-g,v) = FrFunctionalFAF(f,v) - FrFunctionalFAF(g,v)
  proof
    let V, W be non empty ModuleStr over INT.Ring, f, g be FrForm of V,W,
    w be Vector of V;
    now
      let v be Vector of W;
      thus (FrFunctionalFAF(f-g,w)).v = (f-g).(w,v) by HTh8
      .= f.(w,v) - g.(w,v) by Def7
      .= (FrFunctionalFAF(f,w)).v - g.(w,v) by HTh8
      .= (FrFunctionalFAF(f,w)).v - (FrFunctionalFAF(g,w)).v by HTh8
      .= (FrFunctionalFAF(f,w)).v + (-FrFunctionalFAF(g,w)).v by HDef4
      .= (FrFunctionalFAF(f,w) -FrFunctionalFAF(g,w)).v by HDef3;
    end;
    hence thesis by FUNCT_2:63;
  end;
