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\begin{proposition}\label{cases_exact}
Let $A$ be a set.
$A=A$.
\end{proposition}
\begin{proof}
\begin{byCase}
\caseOf{$A=\emptyset$.}
Follows.
\caseOf{$A=\{A\}$.}
Follows.
\caseOf{$A=A$.}
Follows.
\end{byCase}
\end{proof}
\begin{proposition}\label{by_contradiction_exact}
Let $A$ be a set.
$A=A$.
\end{proposition}
\begin{proof}
Suppose not.
Follows.
\end{proof}
\begin{proposition}\label{arbitrary_contradiction_exact}
Let $A$ be a set.
Suppose $A\neq A$.
Then $A=A$.
\end{proposition}
\begin{proof}
Contradiction.
\end{proof}
\begin{proposition}\label{omitted_case_exact}
Let $A$ be a set.
$A=A$.
\end{proposition}
\begin{proof}
\begin{byCase}
\caseOf{$A=A$.}
Omitted.
\caseOf{$A\neq A$.}
Follows.
\end{byCase}
\end{proof}
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