sitemati typo

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2026-05-19 15:33:17 +02:00
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Let $\phi \lor \psi$ be a tautology in $\mathcal{T}_0$, therefore \[\nu(\phi \lor \psi) = 1_{Pcov(\mathcal{A}_i(\mathcal{T}_0))} = [(1_{Set}, 1_{\mathcal{A}_i(\mathcal{T}_0)},un)]
\]
Now, since $\nu (\phi \lor \psi)= \nu(\phi) \oplus \nu(\psi)$ there exists a global section $g: 1_{Pcov(\mathcal{A}_i(\mathcal{T}_0))} \to [(X_\phi , [\phi] , e_\phi)] \oplus [(X_\psi , [\psi], e_\psi)]$.
By remark \ref{global section in cover rmk} $g = (g_1,g_2): 1_{Pcov(\mathcal{A}_i(\mathcal{T}_0))} \to \nu(\phi)$ or $g = (g_1,g_2): 1_{Pcov(\mathcal{A}_i(\mathcal{T}_0))} \to \nu(\psi)$. Now, in the first case $e_\phi (g_1(\star)) \in \Gamma [\phi]$ hence $\phi$ is provable in the logic. In the second case $e_\psi (g_1(\star)) \in \Gamma [\psi]$ and so $\psi$ is provable in the logic. Thus the disjunction property is proved and so we are done.$e_\phi (g_1(\star)) \in \Gamma [\phi]$
By remark \ref{global section in cover rmk} $g = (g_1,g_2): 1_{Pcov(\mathcal{A}_i(\mathcal{T}_0))} \to \nu(\phi)$ or $g = (g_1,g_2): 1_{Pcov(\mathcal{A}_i(\mathcal{T}_0))} \to \nu(\psi)$. Now, in the first case $e_\phi (g_1(\star)) \in \Gamma [\phi]$ hence $\phi$ is provable in the logic. In the second case $e_\psi (g_1(\star)) \in \Gamma [\psi]$ and so $\psi$ is provable in the logic. Thus the disjunction property is proved and so we are done.
\end{proof}
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