all $\pi \in {\mathcal{T}}(\overline{12 \cdots k})$, we have $${\mathcal{S}}(\sigma_q \cdot p) = \frac{ \sum_{(\sigma_q \cdot p)\text{-winnable } \pi
\in {\mathfrak{S}}_N} \theta^{c(\pi)} }{ \sum_{(\sigma_q \cdot p)\text{-prefixed } \pi
\in {\mathfrak{S}}_N} \theta^{c(\pi)} } = \frac{ \sum_{p\text{-winnable } \pi \in {\mathfrak{S}}_N}
\theta^{c(\sigma_q \cdot \pi)} }{ \sum_{p\text{-prefixed } \pi \in {\mathfrak{S}}_N} \theta^{c(\sigma_q \cdot \pi)} }$$ which is $\frac{ \theta^{c(q) - c(12 \cdots k)} }{ \theta^{c(q) - c(12 \cdots
k)}} S(p)$ unless $q$ happens to be ineligible and $p = [12 \cdots k]$ in which case $\sigma_q \cdot p = q$ and so ${\mathcal{S}}(q) = 0$.

The probability ${\mathcal{S}}(p^{\circ})$ is an $\oplus$-sum of strike probabilities, say $${\mathcal{S}}(p^{\circ}) = {\mathcal{S}}(r_1) \oplus {\mathcal{S}}(r_2) \oplus \cdots \oplus {\mathcal{S}}(r_n),$$ for some prefixes $r_i \in \bigcup_{j = m+1}^N {\mathfrak{S}}_j$. Then since $\sigma_q$ is a strike-probability preserving bijection, we have $${\mathcal{S}}((\sigma_q \cdot p)^{\circ}) = {\mathcal{S}}(\sigma_q \cdot r_1) \oplus {\mathcal{S}}(\sigma_q \cdot
r_2) \oplus \cdots \oplus {\mathcal{S}}(\sigma_q \cdot r_n)$$ $$= \frac{ \theta^{c(q) - c(12 \cdots k)} }{ \theta^{c(q) - c(12 \cdots
k)}} {\mathcal{S}}(r_1) \oplus \frac{ \theta^{c(q) - c(12 \cdots k)} }{ \theta^{c(q) - c(12
\cdots k)}} {\mathcal{S}}(r_2) \oplus \cdots \oplus \frac{ \theta^{c(q) - c(12 \cdots k)}
}{ \theta^{c(q) - c(12 \cdots k)}} {\mathcal{S}}(r_n)$$ which is $\frac{ \theta^{c(q) - c(12 \cdots k)} }{ \theta^{c(q) 