 To reconstruct Frege’s strategy for proving this theorem, recall that the weak ancestral of the Predecessor relation, i.e., \(\mathit{Precedes}^{+}(x,y)\), can be read as: \(x\) is a member of the predecessor-series ending with \(y\). Frege then considers the concept member of the predecessor-series ending with n, i.e., \([\lambda z \, \mathit{Precedes}^{+}(z,n)]\), where \(n\) is a natural number. Frege then shows, by induction, that every natural number \(n\) precedes the number of the concept member of the predecessor-series ending with n. That is, Frege proves that every natural number has a successor by proving the following Lemma on Successors by induction: