 We have already seen an example of many-one reducibility in the proof of  Rice’s Theorem (3.4).  In particular, the proof showed that the problem of deciding membership in \(K\) can be reduced to that of deciding membership in any non-trivial index set \(I\) in the following sense: for all \(n\), if \(n \in K\) then \(s^2_1(c,n) \in I\). Thus if there were an algorithm for deciding membership in \(I\), we would be able to decide whether \(n \in K\) by using it to test whether \(s^2_1(c,n) \in I\). The function \(s^2_1(c,x)\) (whose computability is given by the s-m-n Theorem) is thus a so-called many-one reduction of \(K\) to \(I\).