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diff --git a/test/phase5/exact-local-function.tex b/test/phase5/exact-local-function.tex
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+\begin{signature}\label{phase5_local_pair}
+    $(x,y)$ is a set.
+\end{signature}
+
+\begin{proposition}\label{phase5_local_function}
+    For all $A$ we have $A = A$.
+\end{proposition}
+\begin{proof}
+    Fix $A$.
+    Let $f(x) = x$ for $x\in A$.
+    Show for all $y$ we have if $y\in A$, then $(y,y)\in f$.
+    \begin{subproof}
+        Fix $y$.
+        Assume $y\in A$.
+        Follows by assumption.
+    \end{subproof}
+    Follows by assumption.
+\end{proof}