So we may assume $\nu\neq \infty.$ Let $\mathcal{F}:= \{a_0+{\mathbf{a}}.{\mathbf{f}}_\nu+\Theta_\nu \ |\  (a_0,{\mathbf{a}})\in\mathcal{O}^{n+1}\}$. This is a compact family of functions of $C^l(U_\nu)$ for every $l>0 $ since $\mathcal{O}$ is compact in ${\mathbb{Q}}_\nu$. Now if this family satisfies condition (\[3.4\]) for some $l\in {\mathbb{N}}$, then the conclusion follows from the previous Proposition. Hence we may assume that the family does not satisfy (\[3.4\]) for every $l\in {\mathbb{N}}$. Then by the continuity of differential and the compactness of $\mathcal{O}$, there exists ${\mathbf{c}}_l\in \mathcal{O}^n$ such that for every $2 \leq l\in {\mathbb{N}}$ we have $$\max_{|\beta|\leq l}|\partial_{ \beta}({\mathbf{c}}_l.f_\nu+\Theta_\nu)({\mathbf{x}}_0)| > 0.$$ Now this sequence $\{{\mathbf{c}}_l\} \in\mathcal{O}^n$ has a convergent subsequence $\{{\mathbf{c}}_{l_k}\}$ converging to ${\mathbf{c}}\in \mathcal{O}^n$ since $\mathcal{O}^n$ is compact. By taking limits we get that $$|\partial_{ \beta}({\mathbf{c}}.f_\nu+\Theta_\nu)({\mathbf{x}}_0)|=0 \ \forall \ \beta.$$ However, as each of the ${\mathbf{f}}_{\nu}$ and $\Theta_\nu$ are analytic on $U_\nu,$ there exists a neighbourhood $V_{{\mathbf{x}}_0}$ of ${\mathbf{x}}_0$ such that $$({\mathbf{c}}.f_\nu+\Theta_\nu)({\mathbf{x}})=u\ \forall \ {\mathbf{x}}\in V_{{\mathbf{x}}_0},$$ where $u \in {\mathbb{Q}}_\nu$ is a constant. Therefore replacing $\Theta_\nu$ by $u-{\mathbf{c}}.{\mathbf{f}}_{\nu},$ we get that $$\mathcal{F}=\{ a_0+u+({\mathbf{a}}-{\mathbf{c}}).{\mathbf{f}}_{\nu} \ | (a_0,{\mathbf{a}})\in \mathcal{O}^{n+1} \}.$$ First consider the case where $|a_0+u| < 2|{\mathbf{a}}-{\mathbf{c}}|,$ then $$\mathcal{F}_1= \left\{\frac{a_0+u}{|{\mathbf{a}}-{\mathbf{c}}|}+\frac{{\mathbf{a}}-{\mathbf{c}}}{|{\mathbf{a}}-{\mathbf{c}}|}.{\mathbf{f}}_\nu |\  (a_0,{\mathbf{a}})\in\mathcal{O}^{n+1}\right\}$$ is compact in $C^l(U_\nu)$ for every $l\in {\mathbb{N}}$. Then by linear independence of $1,f_\nu^{(1)},\cdots,f_\nu^{(n)},$ 