 is non-empty, then there is a function which picks an element \(a_{n+1}\) from the set, using the parameters \(a_1,\ldots,a_{n}\) as arguments. The second Axiom of Choice (AC′) is a kind of second-order choice: We have for all \(a_1,\ldots,a_n\) a function F with the property \(\phi\), so in fact F depends on \(a_1,\ldots, a_n\) and should be denoted \(F_{a_1\ldots a_n}\). What (AC′) says now is that we can collect the functions \(F_{a_1\ldots a_n}\) together to form just one function \(F'\) of higher arity such that 