 Given this proof of the Lemma on Successors, Theorem 5 is not far away. The Lemma on Successors shows that every number precedes some cardinal number of the form \(\#F\). We still have to show that such successor cardinals are natural numbers. That is, it still remains to be shown that if a number \(n\) precedes something \(y\), then \(y\) is a natural number: 