\setlength\subproofhorizspace{1.3em}
\begin{logicproof}{2}
  \exists x \; P(x) \lor \exists x \; Q(x) & \prem\\
  \begin{subproof}
    \exists x \; P(x)		               & $\assum$\\
    \begin{subproof}
      P(x_0)                               & $\assum$ $\freshVar{$x_0$}$\\
      P(x_0) \lor Q(x_0)                   & $\ori3$\\
      \exists x \; (P(x) \lor Q(x))        & $\existi4$
    \end{subproof}
    \exists x \; (P(x) \lor Q(x))          & $\existe2,3-5$
  \end{subproof}
  \begin{subproof}
    \exists x \; Q(x)		               & $\assum$\\
    \begin{subproof}
      Q(x_0)	                           & $\assum$ $\freshVar{$x_0$}$\\
      P(x_0) \lor Q(x_0)                   & $\ori8$\\
      \exists x \; (P(x) \lor Q(x))        & $\existi9$
    \end{subproof}
    \exists x \; (P(x)\lor Q(x))           & $\existe7,8-10$
  \end{subproof}
  \exists x \; (P(x)\lor Q(x))             & $\ore1,2-6,7-11$
\end{logicproof}