reserve GF for Field,
  V for VectSp of GF,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n, m for Nat;
reserve V for finite-dimensional VectSp of GF,
  W, W1, W2 for Subspace of V,
  u, v for Vector of V;

theorem Th29:
  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 MATRLIN:def 1;
  hereby
    consider I being finite Subset of V such that
A2: I is Basis of V by MATRLIN:def 1;
    assume dim V = 0;
    then card I = 0 by A2,Def1;
    then
A3: I = {}(the carrier of V);
    (Omega).V = the ModuleStr of V by VECTSP_4:def 4
      .= Lin(I) by A2,VECTSP_7:def 3
      .= (0).V by A3,VECTSP_7:9;
    hence (Omega).V = (0).V;
  end;
A4: I <> {0.V} by A1,VECTSP_7:3,def 3;
  assume (Omega).V = (0).V;
  then the ModuleStr of V = (0).V by VECTSP_4:def 4;
  then Lin(I) = (0).V by A1,VECTSP_7:def 3;
  then I = {} or I = {0.V} by VECTSP_7:10;
  hence thesis by A1,A4,Def1,CARD_1:27;
end;
