 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 ThnTV4:
  v is non torsion iff Lin{v} is free & v <> 0.V
  proof
    hereby
      assume v is non torsion;
      then {v} is linearly-independent Subset of V by ThnTV3;
      hence Lin{v} is free & v <> 0.V;
    end;
    assume A1: Lin{v} is free & v <> 0.V;
    then A2: Lin{v} is torsion-free;
    A3: v <> 0.Lin{v} by A1,ZMODUL01:26;
    v in {v} by TARSKI:def 1;
    then v in Lin{v} by ZMODUL02:65;
    then reconsider vl = v as Vector of Lin{v};
    vl is non torsion by A2,A3;
    hence v is non torsion by ThTV6;
  end;
