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\label{thm:sheaf-axioms}
The holomorphic presheaf
$\mathcal{O}_\tau$
(Definition~\textup{\ref{def:holomorphic-presheaf}}, II.D47)
is a \textbf{sheaf} on the cylinder topology of~$\tau^3$.
Explicitly:
\begin{enumerate}
    \item[\textup{(S1)}]
          \textbf{Locality.}
          If $f \in \mathcal{O}_\tau(U)$
          restricts to zero on every element
          of a cover $\{U_i\}$ of~$U$,
          then $f = 0$
          \textup{(Proposition~\ref{prop:ch36-locality})}.
    \item[\textup{(S2)}]
          \textbf{Gluing.}
          If $\{f_i \in \mathcal{O}_\tau(U_i)\}$
          are compatible on pairwise overlaps,
          then there exists a unique
          $f \in \mathcal{O}_\tau(U)$
          restricting to~$f_i$ on each~$U_i$
          \textup{(Lemma~\ref{lem:gluing}, II.L06)}.
\end{enumerate}

Locality was proved in Proposition~\ref{prop:ch36-locality}.
Gluing was proved in Lemma~\ref{lem:gluing}.
The presheaf axioms (P1) and (P2)
hold by construction
(Definition~\ref{def:holomorphic-presheaf}).
Together, (P1), (P2), (S1), (S2)
establish that $\mathcal{O}_\tau$ is a sheaf.