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+\begin{proposition}\label{phase5_fixed_set_induction}
+    For all sets $x$ we have $x=x$.
+\end{proposition}
+\begin{proof}
+    Fix $x$.
+    Show $x=x$.
+    \begin{subproof}[Proof by \in-induction on $x$]
+        Follows.
+    \end{subproof}
+    Follows by assumption.
+\end{proof}