every ${\mathbf{x}}\in  U_{{\mathbf{x}}_0}=\bigcap_{i=1}^r U_{({\mathbf{x}}_0f_i)} $ and $f\in\mathcal{F}$ there exists $A_{f_i}$ such that $$\label{final_cond}
              \Phi_1(A_{f_i},f,{\mathbf{x}}) >\frac{C}{2}.$$

Choose $\delta=\frac{C}{4u}$ where $ u $ is the constant coming from the inequality $$|\partial_i^k\Theta\circ T(T{^{\text{-}1}}{\mathbf{x}})| \leq u \max_{|\beta|\leq l}|\partial_{ \beta} f({\mathbf{x}}) |$$ for $T\in {\operatorname{GL}}_d(\mathcal{O})$. Thus any $\Theta $ satisfying (\[theta\_cond\]) will also satisfy $$\Phi_1(A_{f_i},f+\Theta,{\mathbf{x}})>\frac{C}{4} \ \ \forall \ {\mathbf{x}}\in U_{{\mathbf{x}}_0}.$$ By the compactness of $\mathcal{F}$ and (\[theta\_cond\]) there is a uniform upper bound for every $f\in\mathcal{F}$ and $\Theta $ of the aforementioned type. Now applying Theorem \[theorem 3.2\] we have that $f+(\Theta\circ A_{f_i})$ is $(dk^{3-\frac{1}{k}},\frac{1}{dk})$-good on $A_{f_i}^{-1} U_{{\mathbf{x}}_0}$. Therefore, $f+\Theta$ is $(dk^{3-\frac{1}{k}},\frac{1}{dk})$-good on $U_{{\mathbf{x}}_0}$. This completes the proof of the first part.\
Now consider the set $\mathcal{F}_{A_{f_i}}= \{f\in\mathcal{F} \ | \ \Phi_1(A_{f_i},f,{\mathbf{x}}_0)\geq \frac{C}{2}\}$. Clearly this is a closed subset of the compact set $\mathcal{F}$, so it is also compact. Therefore $\{\partial_j(f\circ A_{f_i}) | \  f\in\mathcal{F}_{A_{f_i}} \}$ is also compact being the image of a compact set under a continuous map. Since $\mathcal{F} \subset \bigcup_{i=1,\cdots,r} \mathcal{F}_{A_{f_i}}$, we may, without loss of generality, take the same $A$ for every $f\in \mathcal{F}$. Now we want to apply the first part of this Proposition. Suppose $|\beta| \geq 2 $ in (\[3.6\]), then to apply part(1) we have to 