 The various parts of  Theorem 3.11  follow from prior definitions together with Propositions  3.2  and  3.7.  Note in particular that it follows from parts ii and iv of  Theorem 3.11  together with part vii of  Proposition 3.7  that \(\emptyset^{(n)}\) is an example of a set in the class \(\Sigma^0_n - \Pi^0_n\) from which it also follows that \(\overline{\emptyset^{(n)}} \in \Pi^0_n - \Sigma^0_n\). This observation in turn strengthens the Hierarchy Theorem  (3.10)  by showing that \(\Delta^0_n \subsetneq \Sigma^0_n\) and \(\Delta^0_n \subsetneq \Pi^0_n\) as depicted in Figure 3.