 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
  for V being Z_Module,
      a being Element of INT.Ring,
      v,w being Vector of V holds
  V is Mult-cancelable & a <> 0.INT.Ring & a * v = a * w implies v = w
  proof
    let V be Z_Module,
        a be Element of INT.Ring,
        v,w be Vector of V;
    assume A1:V is Mult-cancelable;
    set R = INT.Ring;
    assume that
    A2: a <> 0.R and
    A3: a * v = a * w;
    0.V = a * v - a * w by A3,RLVECT_1:15
    .= a * (v - w) by Th8;
    then v - w = 0.V by A2,A1;
    hence thesis by RLVECT_1:21;
  end;
