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This sequent is provable.
\begin{logicproof}{2}
\lnot \lnot \lnot p \land q & \prem \\
\lnot \lnot r & \prem \\
\lnot \lnot \lnot p & $\ande{1}1$ \\
q & $\ande{2}1$ \\
r & $\negnege2$ \\
\lnot p & $\negnege3$ \\
\lnot \lnot q & $\negnegi4$ \\
r \land \lnot p & $\andi 5,6$ \\
r \land \lnot p \land \lnot \lnot q & $\andi 8,7$
\end{logicproof}