
theorem Th7:
  for V being RealUnitarySpace, W being Subspace of V st V is
  finite-dimensional holds W is finite-dimensional
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  set A = the Basis of W;
  consider I being Basis of V such that
A1: A c= I by RUSUB_3:24;
  assume V is finite-dimensional;
  then I is finite by Th3;
  hence thesis by A1;
end;
