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Assume the theorem is false. We know the following: 
\begin{equation*} \label{1}
(\forall v_{0})\ \operatorname{FUNCTION}\left(v_{0}\right) \;\wedge\; \operatorname{FUNCTION}\left({{v_{0}}^{-1}}\right) \implies \operatorname{ONEONE}\left(v_{0}\right)
\end{equation*}We then substitute $0$ for $v_{0}$ to obtain 
\begin{equation*} \label{5}
\operatorname{FUNCTION}\left(0\right) \;\wedge\; \operatorname{FUNCTION}\left({{
_{0}}^{-1}}\right) \implies \operatorname{ONEONE}\left(0\right)
\end{equation*}which together with ${{0}^{-1}} = 0$ yields 
\begin{equation*} \label{6}
\operatorname{FUNCTION}\left(0\right) \;\wedge\; \operatorname{FUNCTION}\left(0\right) \implies \operatorname{ONEONE}\left(0\right)
\end{equation*}which reduces to 
\begin{equation*} \label{7}
\operatorname{FUNCTION}\left(0\right) \implies \operatorname{ONEONE}\left(0\right)
\end{equation*}which together with $\operatorname{FUNCTION}\left(0\right)$ yields 
\begin{equation*} \label{8}
\operatorname{ONEONE}\left(0\right)
\end{equation*}but this contradicts $\neg \operatorname{ONEONE}\left(0\right)$.

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