The theorem of the complement for nested sub-Pfaffian sets

Let R be an o-minimal expansion of the real field, and let Lnest(R) be the language consisting of all nested Rolle leaves over R. We call a set nested sub-Pfaffian over R if it is the projection of a positive Boolean combination of definable sets and nested Rolle leaves over R. Assuming that R admits analytic cell decomposition, we prove that the complement of a nested sub-Pfaffian set over R is again a nested sub-Pfaffian set over R. As a …