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\setlength\subproofhorizspace{1.3em}
\begin{logicproof}{1}
P(x) \lor Q(y) &\prem\\
P(x)\imp R(z) &\prem\\
Q(y) \imp R(z) &\prem\\
\begin{subproof}
P(x) &$\assum$\\
R(z) &$\impe2,4$
\end{subproof}
\begin{subproof}
Q(y) &$\assum$\\
R(z) &$\impe3,6$
\end{subproof}
R(z) &$\ore1,4-5,6-7$
\end{logicproof}