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\setlength\subproofhorizspace{1.3em}\begin{logicproof}{2} \exists x \;(P(x) \lor Q(x)) & prem.\\ \begin{subproof} \llap{$x_0\enspace \;$} P(x_0) \lor Q(x_0) & ass.\\ \begin{subproof} P(x_0) & ass.\\ \exists x \; P(x) &$ \exists\mathrm{i} 3$\\ \exists x \; P(x) \lor \exists x \; Q(x) &$ \lor\mathrm{i} 4$ \end{subproof} \begin{subproof} Q(x_0) & ass.\\ \exists x \; Q(x) &$ \exists\mathrm{i} 6$\\ \exists x \; P(x) \lor \exists x \; Q(x) &$ \lor\mathrm{i} 7$ \end{subproof} \exists x \;P(x) \lor \exists x \; Q(x) &$ \lor\mathrm{e} 2,3-8$ \end{subproof} \exists x \;P(x) \lor \exists x \; Q(x) & $\exists\mathrm{e} 1,2-9$\end{logicproof}
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