 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,
      v being Vector of V holds
  V is Mult-cancelable & v + v = 0.V implies v = 0.V
  proof
    let V be Z_Module,
        v be Vector of V;
    assume A1: V is Mult-cancelable;
    assume v + v = 0.V;
    then v = - v by RLVECT_1:def 10;
    hence thesis by A1,Th3;
  end;
