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tiny = to eqto in circle
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UlrikBuchholtz committed Nov 8, 2024
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Expand Up @@ -1565,7 +1565,8 @@ \section{A reinterpretation of the circle}\label{sec:S1isC}
\begin{proof}
Using \cref{cor:fib-vs-path}\ref{conn-fib-vs-path} it suffices to show that
each map induced by $f$ on identity types is an equivalence if and only if the specific map
$\ap f : (x=x) \to (f(x) = f(x))$ is an equivalence. Being an equivalence is a proposition,
$\ap f : (x\eqto x) \to (f(x) \eqto f(x))$ is an equivalence.
Being an equivalence is a proposition,
so the result follows in two easy steps from $X$ being connected,
using \cref{xca:component-connected}.
\end{proof}
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