Restricted Log-Exp-Analytic Power Functions

A preparation theorem for compositions of restricted log-exp-analytic functions and power functions of the form $$h:\mathbb{R}\to \mathbb{R},x\mapsto \{\begin{array}{ll}{x}^r,& x>0,\\ 0,& \text{ else, }\end{array}$$ for $r\in \mathbb{R}$ is given. Consequently we obtain a parametric version of Tamm’s theorem for this class of functions which is indeed a full generalisation of the parametric version of Tamm’s theorem for ${\mathbb{R}}_{\text{an}}^{\mathbb{R}}$-definable functions.