reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;
reserve V,W for finite-rank free Z_Module;
reserve T for linear-transformation of V,W;

theorem ThTFX:
  for V being Z_Module holds
  V is torsion-free iff (Omega).V is torsion-free
  proof
    let V be Z_Module;
    thus V is torsion-free implies (Omega).V is torsion-free;
    assume A1: (Omega).V is torsion-free;
    for v being Vector of V st v <> 0.V holds v is non torsion
    proof
      let v be Vector of V such that
      B1: v <> 0.V;
      reconsider vv = v as Vector of (Omega).V;
      B2: vv is non torsion by A1,B1;
      for i being Element of INT.Ring st i <> 0.INT.Ring holds i * v <> 0.V
      proof
        let i be Element of INT.Ring such that
        C1: i <> 0.INT.Ring;
        i * vv <> 0.(Omega).V by B2,C1;
        hence thesis;
      end;
      hence thesis;
    end;
    hence V is torsion-free;
  end;
