 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 ThTV6:
  for v being Vector of V, w being Vector of W st v = w holds
  v is torsion iff w is torsion
  proof
    let v be Vector of V, w be Vector of W such that
    A1: v = w;
    hereby
      assume v is torsion;
      then consider i be Element of INT.Ring such that
      B2: i <> 0 & i*v = 0.V;
      i*w = i*v by A1,ZMODUL01:29
      .= 0.W by ZMODUL01:26,B2;
      hence w is torsion by B2;
    end;
    assume w is torsion;
    then consider i be Element of INT.Ring such that
    B2: i <> 0 & i*w = 0.W;
    i*v = i*w by A1,ZMODUL01:29
    .= 0.V by ZMODUL01:26,B2;
    hence v is torsion by B2;
  end;
