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Suppose $\lambda \in \partial {\sigma }_B(x)$. If $\lambda \in {\rho }_A(x)$, choose a connected open disc about it contained in ${\rho }_A(x)$. On that disc the clopen subset ${\rho }_B(x)$ is either the whole disc or empty. Neither possibility permits a boundary point of ${\sigma }_B(x)$, so $\lambda \in {\sigma }_A(x)$. But ${\sigma }_A(x)\subseteq {\sigma }_B(x)$, so a point in the interior of ${\sigma }_A(x)$ is in the interior of ${\sigma }_B(x)$. Our point cannot be such an interior point. ConsequentlyThis is boundary inclusion of spectra in a closed unital subalgebra, established from the preceding arguments rather than quoted.