For a fixed $\delta<\delta_{0}$, there is a $k_{1}>k_{0}$ such that, for any $y\in B_{h_{E}}(0,\frac{3r}{2})$ and $k>k_{1}$, there is a unique $\sigma_{k}(y)\in\mathfrak{B}_{1}$, such that

$$\mathfrak{F}_{k}(y,\sigma_{k}(y))=0,\ \ \ \|\sigma_{k}(y)\|_{C^{1,\alpha}(L,h)}\leq\delta,$$

which implies that $L(y,\sigma_{k}(y))$ is a special lagrangian submanifold of $(Y_{2r},\omega_{k},\Omega_{k})$. Furthermore,

$$\|D\sigma_{k}(y)\|\leq 2n\delta\overline{C}C\|(d\alpha_{k},d\beta_{k})\|_{C^{1,\alpha}(Y_{2r},g)},$$

for a constant $C$ independent of $k$.