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\item \self Consider the following natural deduction proof for the sequent $$\quad \exists x \; P(x) \lor \exists x \; Q(x) \quad \ent \quad \exists x \; (P(x)\lor Q(x)).$$ Is the proof correct? If not, explain the error in the proof and either show how to correctly prove the sequent, or give a counterexample that proves the sequent invalid.
\setlength\subproofhorizspace{1.7em} \begin{logicproof}{2} \exists x \; P(x) \lor \exists x \; Q(x) & prem.\\ \begin{subproof} \exists x \; P(x) & ass.\\ \begin{subproof} \llap{$x_0\enspace \;$} P(x_0) & ass.\\ P(x_0) \lor Q(x_0) & $\lor \mathrm{i}_1$ 3 \end{subproof} \exists x \; (P(x) \lor Q(x)) & $\exists \mathrm{e}$ 2,3-4 \end{subproof} \begin{subproof} \exists x \; Q(x) & ass.\\ \begin{subproof} \llap{$x_0\enspace \;$} Q(x_0) & ass.\\ P(x_0) \lor Q(x_0) & $\lor \mathrm{i}_2$ 7 \end{subproof} \exists x \; (P(x)\lor Q(x)) & $\exists \mathrm{e}$ 6,7-8 \end{subproof} \exists x \; (P(x)\lor Q(x)) & $\lor \mathrm{e}$ 1,2-5,6-9 \end{logicproof}
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