 reserve V for Z_Module;
 reserve W for Subspace of V;
 reserve v, u for Vector of V;
 reserve i for Element of INT.Ring;

theorem ThMCTF:
  V is Mult-cancelable iff V is torsion-free
  proof
    hereby
      assume V is Mult-cancelable; then
      for v being Vector of V st v <> 0.V holds v is non torsion;
      hence V is torsion-free;
    end;
    assume AS: V is torsion-free;
    for i being Element of INT.Ring, v being Vector of V holds
    i <> 0.INT.Ring & v <> 0.V implies i * v <> 0.V
    proof
      let i be Element of INT.Ring, v be Vector of V;
      assume A1: i <> 0.INT.Ring & v <> 0.V; then
      v is non torsion by AS;
      hence i * v <> 0.V by A1;
    end;
    hence V is Mult-cancelable;
  end;
