 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 Th9:
  for V being Z_Module,
      a,b being Element of INT.Ring,
      v being Vector of V holds
  (a - b) * v = a * v - b * v
proof
  let V be Z_Module,
      a,b be Element of INT.Ring,
      v be Vector of V;
  thus (a - b) * v = a * v + (- b) * v by VECTSP_1:def 15
    .= a * v + b * (- v) by Th5
    .= a * v - b * v by Th6;
end;
