\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}
