 This clearly has the Zorn property.So taking "Q" a maximal member, if "Q" is a basis for the whole Hilbert space "H", we are done.If this were not the case, then letting formula_41, it is easy to see that this would be an formula_25-invariant non-trivial closed subspace; and thus by the lemma above, therein would lie a common eigenvector for the operators (necessarily orthogonal to "Q").