I can't follow the proof of Claim 2.6 and 2.7 of III: Jonsson algebras in inaccessible cardinals of Shelah's Cardinal arithmetic. The part I can't follow is line 12-20 on page 130:
If $\alpha_{\ell}, k_{\ell}$ satisfy $\otimes_{\ell}$, (so $\alpha \in E^0$) then $\alpha_{\ell} > \gamma$ (as $\gamma \notin C_{k_{\ell}}^{\ell}[E_{k_{\ell}}]$), so $e_{\alpha_{\ell}}$
is well defined and $\alpha_{\ell} = \sup e_{\alpha_{\ell}}$; so for $k \ge k_{\ell}, \epsilon_{\ell} =: \min[e_{\alpha_{\ell}} \backslash \gamma]$ is well
defined. Now for $k \ge k_{\ell}$, (as $\alpha_{\ell} > \gamma > \beta^*$ hence $\alpha_{\ell} > \beta_{\delta^*}^{\ell}[E_k]$) we have
$\epsilon_{\ell} \ge \sup[E_k \cap \alpha_{\ell} \cap C_{\delta^*}^n[E_k]]$ (as $\epsilon_{\ell} \ge \gamma$ and $[\gamma, \alpha_{\ell}$) is disjoint to $C_{\delta^*}^n[E_k]$)
and so, looking at the definition of $C_{\delta^*}^{\ell+1}[E_k]$ with $\delta, n, \alpha(*), \alpha, E$ there
corresponding to $\delta^*, \ell, \alpha_{\ell}, \epsilon_{\ell}, E_k$ here, we get $\sup(\epsilon_{\ell} \cap E_k) \in C_{\delta^*}^{\ell+1}[E_k]$. Now
$\langle \sup(\epsilon_{\ell} \cap E_k) : k_{\ell} \le k < \omega \rangle$ is a non-increasing sequence of ordinals, hence
for some $k_{{\ell}+1}, k_\ell < k_{\ell+1} < \omega$ and for some $\alpha_{\ell+1}, \alpha_{\ell+1} = \sup(\epsilon_{\ell} \cap E_k)$ for
every $k \ge k_{\ell+1}$. So it is not hard to check $\otimes_{\ell+1}$ holds.
By the definition of $C_{\delta^*}^{\ell+1}[E_k]$, it seems to me that when the author claims that $\sup(\epsilon_{\ell} \cap E_{k}) \in C_{\delta^*}^{\ell+1}[E_k]$, he uses the fact that there exists $\beta < \delta$ such that $|\delta| > \sup \{\text{cf}(\alpha) : \beta < \alpha \in C_{\delta^*}^{\ell}[E_k] \}$. But I can't prove it. Could someone help me? Thanks.