In detail, we have $U\sim _{\mathcal{P}\webleft (R\webright )}V$ iff the following equivalent conditions hold:
- We have $\chi _{\text{pt}}\subset V\mathbin {\diamond }R\mathbin {\diamond }U$.
- We have $\webleft (V\mathbin {\diamond }R\mathbin {\diamond }U\webright )^{\star }_{\star }=\mathsf{true}$, i.e. we have
\[ \int ^{a\in A}\int ^{b\in B}V^{\star }_{b}\times R^{b}_{a}\times U^{a}_{\star }=\mathsf{true}. \]
- There exists some $a\in A$ and some $b\in B$ such that:
- We have $U^{a}_{\star }=\mathsf{true}$.
- We have $R^{b}_{a}=\mathsf{true}$.
- We have $V^{\star }_{b}=\mathsf{true}$.
- There exists some $a\in A$ and some $b\in B$ such that:
- We have $a\in U$.
- We have $a\sim _{R}b$.
- We have $b\in V$.
