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 Th27:
  dim V = dim (Omega).V
proof
  consider I being finite Subset of V such that
A1: I is Basis of V by MATRLIN:def 1;
A2: (Omega).V = the ModuleStr of V by VECTSP_4:def 4
    .= Lin(I) by A1,VECTSP_7:def 3;
  card I = dim V & I is linearly-independent by A1,Def1,VECTSP_7:def 3;
  hence thesis by A2,Th26;
end;
