  In the lecture notes it is shown that an ordinal
$\psi_{\Omega}(\varepsilon_{\mathbb{S}^{+}+1})$ is an upper bound for the
proof-theoretic ordinal of a set theory ${\sf
KP}\omega+(M\prec_{\Sigma_{1}}V)$. In this note we show that ${\sf
KP}\omega+(M\prec_{\Sigma_{1}}V)$ proves the well-foundedness up to
$\psi_{\Omega}(\omega_{n}(\mathbb{S}^{+}+1))$ for each $n$.
