$\zeta_1(j),\zeta_2(j)$ of $E_j$ in angle $\tilde\alpha<\frac\pi2$.

[**Proof:** ]{} We follow an argument due to Peter Jones [@garnettjones]. Let $\{h_m\}$ be the functions provided by Lemma \[lem2\], and let us write $U$ as the disjoint union of sets $U_m\subset B(z_m,\frac{1-|z_m|}{4A^2})$, and let us define $$F(\zeta) = \sum_m\frac1\pi\iint_{U_m}\frac{h_m(\zeta)}{h_m(z)}\frac{b(z)}{\zeta-z}\,dx\,dy.$$ Formally, $\bar\partial F(\zeta)=b(\zeta)$, so it suffices to check the convergence of the sum. Noting that $|h_m(\zeta)|\ge\frac12$ in $U_m$ by Schwarz’s lemma, termwise estimates give $$\begin{aligned}
    \left|\frac1\pi\iint_{U_m}\frac{h_m(\zeta)}{h_m(z)}\frac{b(z)}{\zeta-z}\,dx\,dy\right|
        &\le& \frac2\pi|h_m(\zeta)|\iint_{U_m}\frac{|b(z)|}{|\zeta-z|}\,dx\,dy  \\
        &\le& \frac1\pi\Vert B\Vert_\infty|h_m(\zeta)|\frac{(1-|z_m|)^{-1}}{1-(4A^2)^{-1}}
            \iint_{B\left(z_m,\frac{1-|z_m|}{4A^2}\right)}\frac{dx\,dy}{|\zeta-z|}     \\
    \left|\frac1\pi\iint_{U_m}\frac{h_m(\zeta)}{h_m(z)}\frac{b(z)}{\zeta-z}\,dx\,dy\right|
        &\le& \frac4{3A^2}|h_m(\zeta)|\Vert B\Vert_\infty.
    \end{aligned}$$ Summing, we find that $$\Vert F\Vert_\infty \le \frac{16}3\frac{\log4A^2}{\log\beta^{-1}}\Vert B\Vert_\infty$$ and $$|F(\zeta)|\le\frac{16}3\frac{\log4A^2}{\log\beta^{-1}}n^{-6}\Vert B\Vert_\infty
        \left(\frac{d(\zeta,\{\zeta_1,\zeta_2\})}{{{\rm{diam}}}(L_j)}\right)^2,$$ for $\zeta\in L_j$ due to the special properties of the functions $h_m$.\
Given this, the functions $$G_\mu(z)=\tilde g_\mu(z)+\sum_{\nu=1}^M(a_{\mu\nu}(z)-a_{\nu\mu}(z))f_\nu(z),$$ are corona solutions in $\mathbb D$ satisfying $$|G_\mu(z)-G_\mu^j(z)| \le \sum_{\nu=1}^M|f_\nu(z)||a_{\mu\nu}(z)-a_{\nu\mu}(z)|\lesssim M n^{-6}
        \left(\frac{d(z,\{\zeta_1,\zeta_2\})}{{{\rm{diam}}}(L_j)}\right)^2$$ on $L_j$. For fixed $\tilde\alpha$, if the angle $\alpha$ defining the lozenges $L^n_{J,j}$ in (\[lozenges\]) is sufficiently small then $\phi^n_J(L^n_{J,j})\subset L_j$ (reindexing as appropriate). Mapping back to the domains $\tilde T^n_J$, since $(\phi^n_J)^{-1}$ behaves as $(z-\zeta_i(j))^{\frac{1+\alpha}2}$ about $\zeta_i(j)$, and so we obtain $$\begin{aligned}
    |g^{(n,J)}_\mu(z)-g^j_\mu(z)| &\lesssim& n^{-6}\left(\frac{d(z,\{x_1,x_2\})}{{{\rm{diam}}}(L^n_{J,j})}\right)^{1+\alpha} \\
        &\lesssim& n^{-6}\frac{d(z,K)}{{{\rm{diam}}}(L^n_{J,j})},
    \end{aligned}$$ on $L^n_{J,j}$, where $\{x_1,x_2\}=K\cap\overline{L^n_{J,j}}$, and $\Vert g^{(n,J)}_\mu\Vert_\infty\le C(M,\eta)$, with constants uniform in $(n,J)$.

This completes the proof of the proposition.\

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