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This sequent is provable.
\begin{logicproof}{1}
p \lor \lnot \lnot q & \prem \\
\lnot p \land \lnot q & \prem \\
\begin{subproof}
p & \assum \\
\lnot p & $\ande{1} 2$ \\
\bot & $\nege 3,4$ \\
s \lor \lnot t & $\bote 5$
\end{subproof}
\begin{subproof}
\lnot \lnot q & \assum \\
\lnot q & $\ande{2} 2$ \\
\bot & $\nege 7,8$\\
s \lor \lnot t & $\bote 9$
\end{subproof}
s \lor \lnot t & $\ore 1, 3-6, 7-10$
\end{logicproof}