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Let $S(\Phi,U,K,K_{U})$ denote the solution of the master equation $(S,S)=0$, where $\{\Phi ^{A},U\}$ are the fields and $\{K_{A},K_{U}\}$ are the sources coupled to the $\Phi ^{A}$- and $U$-gauge transformations. If we replace $U$ with the solution $U^{*}(\Phi ,K,K_{U})$ of the $U$-field equations
\begin{equation}
\frac{\delta _{r}S}{\delta U}=0,
\end{equation}
then the action
\begin{equation}
S^{*}(\Phi ,K,K_{U})=S(\Phi ,U^{*}(\Phi ,K,K_{U}),K,K_{U})
\end{equation}
satisfies the master equation $(S^{*},S^{*})=0$ in the reduced set of fields and sources $\Phi,K$.

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