For any $\delta_{0}\ll 1$, there is a constant $k_{0}\gg 1$ such that, if $\|y\|_{h_{E}}\leq\frac{3r}{2}$ and $\|\sigma\|_{C^{1,\alpha}(L,h)}\leq\delta_{0}$, and $k>k_{0}$, then

$$\|D_{\sigma}\mathfrak{F}_{k}(y,\sigma)-D_{\sigma}\mathfrak{F}_{k}(0,0)\|\leq\frac{1}{2\overline{C}}.$$

Furthermore, $D_{\sigma}\mathfrak{F}_{k}(y,\sigma)$ is also invertible, and

$$\|D_{\sigma}\mathfrak{F}_{k}(y,\sigma)^{-1}\|\leq 2\overline{C}.$$