 This scheme may be justified as follows. A proof of the premise should provide a method that given \(m\) provides a number \(n\) such that \(mRn\). Thus the function \(\alpha\) on the natural numbers \(\mathbb{N}\) can be constructed step-by-step: first an element \(m_0\) is chosen such that \(0Rm_0\), which will be the value of \(\alpha(0)\). Then an element \(m_1\) is chosen such that \(1Rm_1\), which will be the value of \(\alpha(1)\), and so on. 