Mild parametrizations of power-subanalytic sets

We obtain two uniform parametrization theorems for families of bounded sets definable in ${\mathbb{R}}_{an}^{\mathbb{R}}$. Let $X = {X_t \subset (0,1)^n \mid t \in T}$beadefinablefamilyofsets$X_t$ofdimensionatmost$m$. Firstly, $X_t$ admits a $C^r$-parametrization consisting of $c{r}^m$ maps for some positive constant $c=c(X)$, which is uniform in $t$. Secondly, $X_t$ admits a $C$-mild parametrization for any $C>1$, which is also uniform in $t$.