Fix a $\delta<\delta_{0}$, there is a $k_{1}>k_{0}$ such that, for $k>k_{1}$, and any $y\in B_{h_{E}}(0,2r)$,

$$\|\mathfrak{F}_{k}(y,0)\|_{C^{0,\alpha}(L,h)}\leq\frac{\delta}{4\overline{C}},$$

by Lemma 4.2. By Theorem 2.3, Lemma 4.3 and 4.4, 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

(9)

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

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

By (6) (7) and (8),

	$\displaystyle\|D_{y}\mathfrak{F}_{k}(y,\sigma_{k})\|$	$\displaystyle\leq$	$\displaystyle C\|(d\alpha_{k},d\beta_{k})\|_{C^{1,\alpha}(Y_{2r},g)}(\sum_{l=0,1,\cdots,n}\|\sigma_{k}\|_{C^{1,\alpha}(L,h)}^{l})$
		$\displaystyle\leq$	$\displaystyle C\|(d\alpha_{k},d\beta_{k})\|_{C^{1,\alpha}(Y_{2r},g)}n\delta,$

for a constant $C$ independent of $k$. By Theorem 2.3,

$$D\sigma_{k}(y)\dot{y}=-D_{\sigma}\mathfrak{F}_{k}(y,\sigma_{k})^{-1}D_{y}\mathfrak{F}_{k}(y,\sigma_{k})\dot{y}.$$

We obtain the conclusion from Lemma 4.4. ∎