\frac{z^Y_1}{z^X_1+z^Y_1} 

        \frac{z^X_1}{z^X_1+z^Y_2} 

        \frac{z^X_2}{z^X_2+z^Y_2} 

        \cdots

        \P^{G\setminus Y}_S(\xi^{G\setminus Y}) \; \P^{G\setminus X}_S(\xi^{G\setminus X})

        \tag{definition} \\

        \tag{definition of $p_{i,j}$} \\

        &= \P(\text{path in grid}) \; \P^{G\setminus Y}_S(\xi^{G\setminus Y}) \; \P^{G\setminus X}_S(\xi^{G\setminus X})

    \end{align*}

    In the grid we see that at every point the probabilities sum to 1, and we always reach the end, so we know the sum of all paths in the grid is 1. This proves the required equality.

    We obtain

    \begin{align*}