reserve V for RealLinearSpace,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n for Element of NAT,
  v for VECTOR of V,
  KL1, KL2 for Linear_Combination of V,
  X for Subset of V;
reserve V for finite-dimensional RealLinearSpace,
  W, W1, W2 for Subspace of V,
  u, v for VECTOR of V;

theorem Th32:
  dim V = 0 iff (Omega).V = (0).V
proof
  consider I being finite Subset of V such that
A1: I is Basis of V by Def1;
  hereby
    consider I being finite Subset of V such that
A2: I is Basis of V by Def1;
    assume dim V = 0;
    then card I = 0 by A2,Def2;
    then
A3: I = {}(the carrier of V);
    (Omega).V = the RLSStruct of V by RLSUB_1:def 4
      .= Lin(I) by A2,RLVECT_3:def 3
      .= (0).V by A3,RLVECT_3:16;
    hence (Omega).V = (0).V;
  end;
A4: I <> {0.V}  by A1,RLVECT_3:8,def 3;
  assume (Omega).V = (0).V;
  then the RLSStruct of V = (0).V by RLSUB_1:def 4;
  then Lin(I) = (0).V by A1,RLVECT_3:def 3;
  then I = {} or I = {0.V} by RLVECT_3:17;
  hence thesis by A1,A4,Def2,CARD_1:27;
end;
