Proposition 3.4 in [@BKM]. Here $\nu=\infty$ is exactly Proposition 1 of [@BaBeVe] so we assume that $\nu\neq\infty$. By (\[3.4\]) there exists $C_1 > 0$ such that for any $f\in \mathcal{F}$ there exists a multiindex $\beta$ with $0<|\beta|=k\leq l $ , where $k=k(f)$ such that $$\label{3.6}
                |\partial_{\beta} f ({\mathbf{x}}_0)|\geq C_1.$$ By the compactness of $\mathcal{F}$, $\inf_{f\in\mathcal{F}}\max_{|\beta|\leq l} \ |\partial_{\beta}f({\mathbf{x}}_0)|$ will be actually attained for some f and we may take that value to be $C_1$. Since there are finitely many $\beta$, we can consider the subfamily $\mathcal{F}_\beta:=\{f\in\mathcal{F}\ |\ \partial_{\beta} f ({\mathbf{x}}_0)|\geq C_1\} $, which is also compact in $C^l(U)$ and satisfies (\[3.4\]). Proving the theorem for $\mathcal{F}_\beta$ will yield sets $U_\beta$ where (1) and (2) above hold. Setting $V_{\nu} := \bigcap_{\beta} U_{\beta}$ then proves the Proposition. We may therefore assume without loss of generality that $\beta$ is the same for every $f\in\mathcal{F}$.\
We wish to apply Theorem 3.2 of [@KT] and to do so we need to satisfy (\[eqn 3.3\]). We are going to show that there exists $A\in {\operatorname{GL}}_d(\mathcal{O})$ such that $f\circ A$ has the property (\[eqn 3.3\]). For $A\in {\operatorname{GL}}_d(\mathcal{O})$ we have, by the chain rule that $$\label{lin_sys1}
                \begin{array}{rcr}
                \partial_{1}^{k}f\circ A(A {^{\text{-}1}}{\mathbf{x}}_0) &=& \sum_{\sum i_j=k, i_j\geq 0} C_{(i_1,\cdots,i_d)} a_{11}^{i_1}\cdots 