 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 & u is torsion implies v - u is torsion
  proof
    assume v is torsion & u is torsion;
    then consider i1, i2 be Element of INT.Ring such that
    A1: i1 <> 0 & i1 * v = 0.V and
    A2: i2 <> 0 & i2 * u = 0.V;
    (i1*i2)*(v-u) = (i1*i2)*v - (i1*i2)*u by ZMODUL01:8
    .= (i2*i1)*v - i1*(i2*u) by VECTSP_1:def 16
    .= i2*(i1*v) - i1*(i2*u) by VECTSP_1:def 16
    .= 0.V - i1*0.V by ZMODUL01:1,A1,A2
    .= 0.V - 0.V by ZMODUL01:1
    .= 0.V;
    hence thesis by A1,A2;
  end;
