 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 ThnTV3:
  v is non torsion iff {v} is linearly-independent
  proof
    A1: not v is torsion implies {v} is linearly-independent
    proof
      assume B1: not v is torsion;
      let l be Linear_Combination of {v};
      assume B4: Sum(l) = 0.V;
      now
        per cases by ZFMISC_1:33,VECTSP_6:def 4;
        suppose
          Carrier(l) = {};
          hence thesis;
        end;
        suppose
          B5: Carrier(l) = {v}; then
          B6: 0.V = l.v * v by B4,ZMODUL02:24;
          now
            assume v in Carrier(l);
            then ex u st v = u & l.u <> 0.INT.Ring;
            hence contradiction by B1,B6;
          end;
          hence thesis by B5,TARSKI:def 1;
        end;
      end;
      hence thesis;
    end;
    {v} is linearly-independent implies not v is torsion
    proof
      assume B1: {v} is linearly-independent;
      assume v is torsion; then
      consider i be Element of INT.Ring such that
      C2: i <> 0 & i*v = 0.V;
      consider l be Linear_Combination of V such that
      C3: l.v = 1 & for u being Vector of V st v <> u holds l.u = 0.INT.Ring
      by ZMODUL03:1;
      for u being Vector of V st u <> v holds not u in Carrier(l)
      proof
        let u be Vector of V;
        assume u <> v;
        then l.u = 0 by C3;
        hence not u in Carrier(l) by ZMODUL02:8;
      end;
      then for u being object holds u = v iff u in Carrier(l) by C3;
      then C5: Carrier(l) = {v} by TARSKI:def 1;
      then C6: Carrier(i*l) = {v} by C2,ZMODUL02:29;
      C7: Carrier(i*l) <> {} by C2,C5,ZMODUL02:29;
      reconsider li = i*l as Linear_Combination of {v}
        by C6,VECTSP_6:def 4;
      Sum(i*l) = i*Sum(l) by ZMODUL02:53
      .= i*((1.INT.Ring)*v) by C3,C5,ZMODUL02:24
      .= 0.V by C2;
      then Sum(li) = 0.V;
      hence contradiction by B1,C7;
    end;
    hence thesis by A1;
  end;
