 Here the function \(s^m_n(i,\vec{x})\) should be thought of as acting on an index \(i\) for an \(n+m\)-ary partial computable function together with values \(\vec{x}\) for the first \(m\) of its arguments. This function returns an index for another partial computable function which computes the \(n\)-ary function determined by carrying out \(\phi^{n+m}_i\) with the first \(m\) of its arguments \(\vec{x}\) fixed but retaining the next \(n\) variables \(\vec{y}\) as inputs. Although the formulation of the s-m-n Theorem may at first appear technical, its use will be illustrated in the proof of  Rice’s Theorem (3.4)  and the  Recursion Theorem (3.5)  below.