On preparation theorems for R<sub>an, exp</sub>-definable functions

In this article we give strong versions for preparation theorems $\mathbb{R}_{\textnormal{an,exp}}$-definable functions outgoing from methods of Lion and Rolin ($\mathbb{R}_{\textnormal{an,exp}}$ is the o-minimal structure generated by all restricted analytic global exponential function). By a deep model theoretic fact Van den Dries, Macintyre Marker every function piecewise given $\mathcal{L}_{\textnormal{an}}(\exp,\log)$-terms where $\mathcal{L}_{\textnormal{an}}(\exp,\log)$ denotes language ordered rings augmented functions, logarithm. So our idea to consider log-analytic at first, i.e. which are iterated compositions either side globally subanalytic logarithm, then as exponential.