, there exists a sequence $\langle \eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$, and $$J_{\alpha}\in \left[0^{\delta_{\eta_\alpha+0}\setminus \delta_{\eta_\alpha}}\right]^{\leq \left|\eta_\alpha\right|},$$ for all $\alpha<\kappa$ such that $$A\subseteq \left\{x\in 0^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{\eta_\alpha+0}\setminus \delta_{\eta_\alpha})\in J_\alpha \right\}.$$  