check condition (\[3.4\]) for the set $\{\partial_j(f\circ A) | \  f\in\mathcal{F} \}$, where we know that $\Phi_1(A,f,{\mathbf{x}}_0)\geq \frac{C}{2}$. Suppose $$\inf_{f\in\mathcal{F}}\max_{|\beta|\leq l-1}|\partial_{ \beta}\partial_{j}(f\circ A)(A{^{\text{-}1}}({\mathbf{x}}_0))|=0.$$ Then by compactness of $\mathcal{F}$ we have that for some $f\in\mathcal{F}$, $$\max_{|\beta|\leq l-1}|\partial_{ \beta}\partial_{j}(f\circ A)(A{^{\text{-}1}}({\mathbf{x}}_0))|=0,$$ which implies that $\Phi_1(A,f,{\mathbf{x}}_0)=0,$ which is a contradiction. Thus by applying the first part of the Proposition we get that for every $j=1,\cdots,d , \partial_j((f+\Theta)\circ A)$ is $(C_\star,\frac{1}{d(l-1)})$-good on an open neighbourhood $B_{A{^{\text{-}1}}({\mathbf{x}}_0)}$ of $A{^{\text{-}1}}({\mathbf{x}}_0)$. So $(\partial_j(f+\Theta\circ A))\circ A{^{\text{-}1}}$ is $(C_\star,\frac{1}{d(l-1)})$-good on $A(B_{A{^{\text{-}1}}({\mathbf{x}}_0)})$. Therefore each $\partial_j(f+\Theta)$ is $(C_\star,\frac{1}{d(l-1)})$-good on $A(B_{A{^{\text{-}1}}({\mathbf{x}}_0)})$ and so is $|\nabla (f+\Theta)|$. The case $|\beta|=1$ in (\[3.6\]) is trivial (See property (G3) of $(C,\alpha)$-good functions). This completes the proof.

As a Corollary, we have,

\[good\_corollary\] Let $U_\nu$ be an open subset of ${\mathbb{Q}}_\nu^{d\nu}, {\mathbf{x}}_0\in U_\nu$ be fixed and assume that ${\mathbf{f}}_\nu=(f_\nu^{(1)},f_\nu^{(2)},\dots,f_\nu^{(n)}): 
    U_\nu\to {\mathbb{Q}}_\nu^n $ satisfies (I2) and (I3) and that $\Theta_\nu $ satisfies (I5). Then there exists a neighbourhood $V_\nu\subset U_\nu$ of ${\mathbf{x}}_0$ and positive constants $ C > 0 $ and $l\in {\mathbb{N}}$ such that for any $(a_0,{\mathbf{a}})\in \mathcal{O}^{n+1},$

1.  $a_0+{\mathbf{a}}.{\mathbf{f}}_{\nu}+\Theta_\nu$ is $(C,\frac{1}{d_\nu l})$-good on $V_\nu,$ and

2.  $|\nabla({\mathbf{a}}.{\mathbf{f}}_\nu +\Theta_\nu)| $ is $ (C,\frac{1}{d_\nu(l-1)})$-good on $V_\nu$.

For the case $\nu=\infty$, see Corollary $3$ of [@BaBeVe] and also [@BKM]. 