reserve V,W for Z_Module;

theorem
  for V being finite-rank free Z_Module holds
  rank V = 1 iff ex v being VECTOR of V st v <> 0.V & (Omega).V = Lin{v}
  proof
    let V be finite-rank free Z_Module;
    hereby
      consider I being finite Subset of V such that
      A1: I is Basis of V by ZMODUL03:def 3;
      assume rank V = 1;
      then card I = 1 by A1,ZMODUL03:def 5;
      then consider v being object such that
  A2: I = {v} by CARD_2:42;
      v in I by A2,TARSKI:def 1;
      then reconsider v as VECTOR of V;
  A3: v <> 0.V by A1,A2,VECTSP_7:def 3;
      Lin{v} = the ModuleStr of V by A1,A2,VECTSP_7:def 3;
      hence ex v being VECTOR of V st v <> 0.V & (Omega).V = Lin{v} by A3;
    end;
    given v being VECTOR of V such that
    A4: v <> 0.V & (Omega).V = Lin{v};
    {v} is linearly-independent & Lin{v} = the ModuleStr of V
    by A4,ZMODUL02:59; then
    A5: {v} is Basis of V by VECTSP_7:def 3;
    card {v} = 1 by CARD_1:30;
    hence thesis by A5,ZMODUL03:def 5;
  end;
