ptstyle \{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
 {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}.\end{aligned}$$ Using [(\[eq:ZA-RCrepr1\])]{}–[(\[eq:ZA-RCrepr2\])]{} to omit “${{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0$” and replace $w_\Lambda({{\bf m}})$ by $w_{{{\cal A}}{^{\rm c}}}({{\bf m}})$, we arrive at [(\[eq:lmm-through\])]{}. This completes the proof of Proposition \[prp:through\].

We explain the meaning of the identity [(\[eq:switching\])]{} and the idea of its proof. Given ${{\bf N}}=\{N_b\}_{b\in{{\mathbb B}}_\Lambda}$, we denote by ${{\mathbb G}}_{{\bf N}}$ the graph consisting of $N_b$ *labeled* edges between ${\underline{b}}$ and ${\overline{b}}$ for every $b\in{{\mathbb B}}_\Lambda$ (see Figure \[fig:switching\]).

$$\begin{aligned}
{{\bf N}}~:&\qquad\includegraphics[scale=0.33]{switching1}\\[5pt]
{{\mathbb G}}_{{\bf N}}~:&\qquad\raisebox{-1.8pc}{\includegraphics[scale=0.33]
 {switching2}}\\[1pc]
{{\mathbb S}}~:&\qquad\raisebox{-1.8pc}{\includegraphics[scale=0.33]
 {switching3}}\\[7pt]
{{\mathbb S}}{\,\triangle\,}\omega~:&\qquad\raisebox{-1.8pc}{\includegraphics[scale=0.33]
 {switching4}}\end{aligned}$$

For a subgraph ${{\mathbb S}}\subset{{\mathbb G}}_{{\bf N}}$, we denote by ${\partial}{{\mathbb S}}$ the set of vertices at which the number of incident edges in ${{\mathbb S}}$ is *odd*, and let ${{\mathbb S}}_{{\cal A}}={{\mathbb S}}\cap{{\mathbb G}}_{{{\bf N}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}}$. Then, the left-hand side of [(\[eq:switching\])]{} equals the cardinality $|{\mathfrak{S}}|$ of $$\begin{aligned}
{\label{eq:Sbefore}}
{\mathfrak{S}}=\{{{\mathbb S}}\subset{{\mathbb G}}_{{\bf N}}:{\partial}{{\mathbb S}}=\{v,x\},~{{\mathbb S}}_{{\cal A}}={\varnothing}\},\end{aligned}$$ and the sum in the right-hand side of [(\[eq:switching\])]{} equals the cardinality $|{\mathfrak{S}}'|$ of $$\begin{aligned}
{\mathfrak{S}}'=\{{{\mathbb S}}\subset{{\mathbb G}}_{{\bf N}}:{\partial