 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
  v is torsion implies i * v is torsion
  proof
    assume v is torsion;
    then consider i1 be Element of INT.Ring such that
    A1: i1 <> 0 & i1 * v = 0.V;
    i1*(i*v) = (i1*i)*v by VECTSP_1:def 16
    .= i*(i1*v) by VECTSP_1:def 16
    .= 0.V by ZMODUL01:1,A1;
    hence thesis by A1;
  end;
