Kleene splits his definition of the unbounded μ-operator to handle the two situations (a) and (b). For situation (b), before the predicate R(x, "y") can serve in an arithmetic capacity in the product Π, its output {t, f} must first be "operated on" by its "representing function χ" to yield {0, 1}. And for situation (a) if one definition is to be used then the "number theoretic function χ" must produce zero to "satisfy" the μ-operator. With this matter settled, he demonstrates with single "Proof III" that either types (a) or (b) together with the five primitive recursive operators yield the (total) recursive functions, with this proviso for a total function: