 reserve R for Ring;
 reserve x, y, y1 for set;
 reserve a, b for Element of R;
 reserve V for LeftMod of R;
 reserve v, w for Vector of V;

theorem Th5:
  for V being Z_Module,
      a being Element of INT.Ring,
      v being Vector of V holds
  a * (- v) = (- a) * v
  proof
    let V be Z_Module,
      a be Element of INT.Ring,
      v be Vector of V;
    thus a * (- v) = a * ((- 1.INT.Ring) * v) by VECTSP_1:14
    .= (a * (- 1.INT.Ring)) * v by VECTSP_1:def 16
    .= (- a) * v;
  end;
