}{{\mathbb S}}={\varnothing},~{{\mathbb S}}_{{\cal A}}={\varnothing}\}.\end{aligned}$$ We note that $|{\mathfrak{S}}|$ is zero when there are no paths on ${{\mathbb G}}_{{\bf N}}$ between $v$ and $x$ consisting of edges whose endvertices are both in ${{\cal A}}{^{\rm c}}$, while $|{\mathfrak{S}}'|$ may not be zero. The identity [(\[eq:switching\])]{} reads that $|{\mathfrak{S}}|$ equals $|{\mathfrak{S}}'|$ if we compensate for this discrepancy.

Suppose that there is a path (i.e., a ) $\omega$ from $v$ to $x$ consisting of edges in ${{\mathbb G}}_{{\bf N}}$ whose endvertices are both in ${{\cal A}}{^{\rm c}}$. Then, the map $$\begin{aligned}
{\label{eq:bijection}}
{{\mathbb S}}\in{\mathfrak{S}}~\mapsto~{{\mathbb S}}{\,\triangle\,}\omega\in{\mathfrak{S}}',\end{aligned}$$ is a bijection [@a82; @ghs70], and therefore $|{\mathfrak{S}}|=|{\mathfrak{S}}'|$. Here and in the rest of the paper, the symmetric difference between graphs is only in terms of *edges*. For example, ${{\mathbb S}}{\,\triangle\,}\omega$ is the result of adding or deleting edges (not vertices) contained in $\omega$. This completes the proof of [(\[eq:switching\])]{}.

We now start with the second stage of the expansion by using Proposition \[prp:through\] and applying inclusion-exclusion as in the first stage of the expansion in Section \[sss:1stexp\]. First, we decompose the indicator in [(\[eq:lmm-through\])]{} into two parts depending on whether or not there is a pivotal bond $b$ for $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
 {\overset{}{\longleftrightarrow}}}x$ from $v$ such that $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
 {\overset{{{\cal A}}}{\longleftrightarrow}}}{\underline{b}}$. Let $$\begin{aligned}
{\label{eq:E-def}}
E_{{\bf N}}(v,x;{{\cal A}})=\{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
 {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\cap\{\nexists
 \text{ pivotal bond }b\text{ for }v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
 {\overset{}{\l