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;

theorem LmTP1:
  for V being Z_Module, v being Vector of V holds
  v is torsion iff v in torsion_part(V)
  proof
    let V be Z_Module, v be Vector of V;
    set W = torsion_part(V);
    hereby
      assume v is torsion;
      then v in { v where v is Vector of V : v is torsion };
      hence v in W by defTorsionPart;
    end;
    assume v in W;
    then v in { v where v is Vector of V : v is torsion } by defTorsionPart;
    then consider vv be Vector of V such that
    A2: vv = v & vv is torsion;
    thus v is torsion by A2;
  end;
