 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 Th3:
  for V being Z_Module,
      v being Vector of V holds
  V is Mult-cancelable & v = - v implies v = 0.V
  proof
    let V be Z_Module,
        v be Vector of V;
    assume A1:V is Mult-cancelable;
    set a = 1.INT.Ring;
    assume v = - v; then
    0.V = v + v by RLVECT_1:def 10
    .= a * v + v
    .= a * v + a * v
    .= (a + a) * v by VECTSP_1:def 15;
    hence thesis by A1;
  end;
