reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;

theorem
  for V be finite-dimensional VectSp of K, B be OrdBasis of (Omega).V
  holds B is OrdBasis of V
proof
  let V be finite-dimensional VectSp of K, B be OrdBasis of (Omega).V;
  reconsider r=rng B as Basis of (Omega).V by MATRLIN:def 2;
  r is linearly-independent by VECTSP_7:def 3;
  then reconsider R=r as linearly-independent Subset of V by VECTSP_9:11;
  Lin R = Lin r by VECTSP_9:17
    .= the ModuleStr of V by VECTSP_7:def 3;
  then R is Basis of V by VECTSP_7:def 3;
  hence thesis by MATRLIN:def 2;
end;
