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This sequent is provable.
\begin{logicproof}{2}
p \land q & \prem \\
q \imp \lnot \lnot r & \prem \\
q & $\ande{2} 1$\\
\lnot \lnot r & $\impe 2,3$ \\
r & $\negnege 4$ \\
p & $\ande{1} 1$ \\
p \land r & $\andi 6,5$
\end{logicproof}