
theorem
  for V being RealUnitarySpace, W being Subspace of V, A being Basis of
  W ex B being Basis of V st A c= B
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let A be Basis of W;
  A is linearly-independent by Def2;
  then reconsider B = A as linearly-independent Subset of V by Th22;
  consider I being Basis of V such that
A1: B c= I by Th15;
  take I;
  thus thesis by A1;
end;
