 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,b being Element of INT.Ring,
      v being Vector of V holds
  V is Mult-cancelable & v <> 0.V & a * v = b * v implies a = b
  proof
    let V be Z_Module,
      a,b be Element of INT.Ring,
      v be Vector of V;
    assume A1: V is Mult-cancelable;
    assume that
A2: v <> 0.V and
A3: a * v = b * v;
    0.V = a * v - b * v by A3,RLVECT_1:15
    .= (a - b) * v by Th9;
    then (- b) + a = 0 by A2,A1 .= 0.INT.Ring;
    hence thesis;
  end;
