reserve V,W for Z_Module;

theorem RL5Th33:
  for V be finite-rank free Z_Module holds
  rank V = 0 iff (Omega).V = (0).V
  proof
    let V be finite-rank free Z_Module;
    consider I being finite Subset of V such that
    A1: I is Basis of V by ZMODUL03:def 3;
    hereby
      consider I being finite Subset of V such that
      A2: I is Basis of V by ZMODUL03:def 3;
      assume rank V = 0;
      then card I = 0 by A2,ZMODUL03:def 5; then
      A3: I = {}(the carrier of V);
      (Omega).V = Lin(I) by A2,VECTSP_7:def 3
      .= (0).V by A3,ZMODUL02:67;
      hence (Omega).V = (0).V;
    end;
    assume (Omega).V = (0).V;
    then Lin(I) = (0).V by A1,VECTSP_7:def 3;
    then I = {} or I = {0.V} by ZMODUL02:68;
    hence thesis by A1,VECTSP_7:def 3,ZMODUL03:def 5,CARD_1:27;
  end;
