Here are two different versions of S-m-n theorem found in two differet sources. I was wondering if they're equivalent? The first talks about the existence of a number, the second talks about the existence of a function. I'm not sure if there's a way to construct one from the other and vice-versa and how such a construction would work. Any help would be appreciated.
There exists a natural number $s^m_n$ with the following property: for all natural $x,y_1,\dots,y_m,z_1,\ \dots, z_n$, $$[\![ [\![s^m_n ]\!](x,y_1,\dots,y_m) ]\!](z_1,\dots,z_n)=[\![x]\!](y_1,\dots,y_m,z_1,\dots,z_n).$$ (The notation $[\![ a ]\!](b)$, also written as $\varphi_a(b)$, stands for the result of the application of the program with number $a$ (in some computable (initailly fixed) enumeration of programs) to input $b$.)
There is a 1:1 computable function $s_n^m(x,y_1,\dots, y_m)$ such that for all $x,y_1,\dots,y_m, z_1,\dots, z_n,$ $$ [\![ s_n^m(x,y_1,\dots, y_m) ]\!] (z_1,\dots, z_n)=[\![ x]\!] (y_1,\dots, y_m,z_1,\dots, z_n)$$