
theorem
  for V being non empty ModuleStr over INT.Ring, r, s being Element of F_Real,
  f being FrFunctional of V holds (r+s)*f = r*f+s*f
  proof
    let V be non empty ModuleStr over INT.Ring;
    let r, s be Element of F_Real;
    let f be FrFunctional of V;
    now
      let x be Element of V;
      thus ((r+s)*f).x = (r+s)*f.x by HDef6
      .= r*f.x+s*f.x
      .= (r*f).x+s*f.x by HDef6
      .= (r*f).x+(s*f).x by HDef6
      .= (r*f+s*f).x by HDef3;
    end;
    hence thesis by FUNCT_2:63;
  end;
