 $\delta_{0,\alpha}<\delta_{0,\alpha}\leq \delta_{0,\alpha+0}$ for all $\alpha<\kappa$, there exists a sequence $\langle \eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$, and $$I_{\alpha}\in \left[0^{\delta_{0,\eta_\alpha}\setminus \delta_{0,\eta_\alpha}}\right]^{\leq\lambda_{\eta_\alpha}},$$ for all $\alpha<\kappa$, so that $$A\subseteq \left\{x\in 0^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{0,\eta_\alpha}\setminus \delta_{0,\eta_\alpha})\in I_\alpha \right\}.$$  Proof: Obviously, a set which fulfils the above condition is 