 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 ThLin5:
  for R being Ring
  for V being LeftMod of R holds
  Lin{0.V} = (0).V
  proof
    let R be Ring;
    let V be LeftMod of R;
    for x being object holds x in Lin{0.V} iff x in (0).V
    proof
      let x be object;
      hereby
        assume x in Lin{0.V};
        then consider l be Linear_Combination of {0.V} such that
        B2: x = Sum(l) by MOD_3:4;
        Sum(l) = l.(0.V) * 0.V by VECTSP_6:17
        .= 0.V by VECTSP_1:14;
        hence x in (0).V by B2,VECTSP_4:17;
      end;
      assume x in (0).V;
      then x = 0.V by VECTSP_4:35;
      hence x in Lin{0.V} by VECTSP_4:17;
    end;
    then for x being Vector of V holds
    x in Lin{0.V} iff x in (0).V;
    hence thesis by VECTSP_4:30;
  end;
