  In arXiv:2208.12944 it is shown that an ordinal
$\sup_{N<\omega}\psi_{\Omega_{1}}(\varepsilon_{\Omega_{\mathbb{S}+N}+1})$ is an
upper bound for the proof-theoretic ordinal of a set theory ${\sf
KP}\ell^{r}+(M\prec_{\Sigma_{1}}V)$. In this paper we show that a second order
arithmetic $\Sigma^{1-}_{2}\mbox{-CA}+\Pi^{1}_{1}\mbox{-CA}_{0}$ proves the
wellfoundedness up to
$\psi_{\Omega_{1}}(\varepsilon_{\Omega_{\mathbb{S}+N+1}})$ for each $N$. It is
easy to interpret $\Sigma^{1-}_{2}\mbox{-CA}+\Pi^{1}_{1}\mbox{-CA}_{0}$ in
${\sf KP}\ell^{r}+(M\prec_{\Sigma_{1}}V)$.
