 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 Th6:
  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;
    set i = 1.INT.Ring;
    thus a * (- v) = (- a) * v by Th5
    .= ((- i) * a) * v
    .= (- i) * (a * v) by VECTSP_1:def 16
    .= - (a * v) by Th2;
  end;
