Complements of Subanalytic Sets and Existential Formulas for Analytic Functions

We show that the complement of a subanalytic set defined by real analytic functions from any subalgebra closed under differentiation is a subanalytic set defined by the functions from the same subalgebra. This result has an equivalent formulation in logic: Consider an expression built from functions as above using equalities and inequalities as well as existential and universal quantifiers. Such an expression is equivalent to an existential expression involving functions from the same class, provided that the variables approach neither infinity nor the boundary of the domain.