 In other words, this theorem asserts that predecessor is a one-to-one relation on the natural numbers. To prove this theorem, it suffices to prove that predecessor is a one-to-one relation full stop. One can prove that predecessor is one-to-one from Hume’s Principle, with the help of the following Equinumerosity Lemma, the proof of which is rather long and involved. The Equinumerosity Lemma asserts that when \(F\) and \(G\) are equinumerous, \(x\) falls under \(F\), and \( y\) falls under \(G\), then the concept object falling under F other than x is equinumerous to the concept object falling under G other than \(y\). The picture is something like this: 