The Theorem of the Complement for Sub-pfaffian Sets

Let R be an o-minimal expansion of the real field, and let P(R) be its Pfaffian closure. Let L be the language consisting of all Rolle leaves added to R to obtain P(R). We prove that P(R) is model complete in the language L, provided that R admits analytic cell decomposition. We do this by proving a somewhat stronger statement, the theorem of the complement for nested sub-Pfaffian sets over R. As a corollary, we obtain that P(R) is obtained by adding to R all nested Rolle leaves over R, a one-stage process.