Nevertheless, here the full Axiom of Choice, granting existence of a map formula_147 into distinguishable elements (equality on the codomain is decidable), implies that formula_148. So the existence claim of general choice functions with formula_149 is non-constructive. To better understand this phenomenon, take note of implications such as formula_150. In this way, the Axiom of Separation ties predicates to set equality, and in turn to information about functions. The difference between the discrete codomain of natural numbers and the domain formula_151 lies in the fact that a priori little is known about the latter. We have formula_152 and formula_153, regardless of formula_27. But in the case of formula_155, there is extensionally only one possible function input to a choice choice function formula_156, so when considering the functional assignment formula_157, then unconditionally declaring formula_158 would not be consistent.