Wellfoundedness proof with the maximal distinguished set

In Arai (An ordinal analysis of a single stable ordinal, submitted) it is shown that an $$\sup _{N<\omega }\psi _{\varOmega _{1}}(\varepsilon _{{\mathbb {S}}+N}+1})$$ upper bound for the proof-theoretic set theory $$\mathsf {KP}\ell ^{r}+(M\prec _{\Sigma _{1}}V)$$ . this paper we show second order arithmetic $$\Sigma ^{1-}_{2}{\mathrm {-CA}}+\Pi ^{1}_{1}{\mathrm {-CA}}_{0}$$ proves wellfoundedness up to $$\psi {S}}+N+1}})$$ each N. It easy interpret in