any family $\{\psi_{i}(\vec{x}) \textrm{ | } i\in I\}$ of geometric formulae in the same context, $\phi \vdash_{\vec{x}} \mathbin{\mathop{\textrm{\huge $\vee$}}\limits_{i\in I}} \psi_{i}$ provable in $\mathbb T$ implies $\phi \vdash_{\vec{x}} \psi_{i}$ provable in $\mathbb T$ for some $i\in I$.

This follows similarly to Theorem \[Onetoposmany\_cartcrit\] by using Lemma \[Onetoposmany\_equiv\](v) and Proposition \[Onetoposmany\_identif\](iv).

\[Onetoposmany\_cohcrit\] Let $\mathbb T$ be a geometric theory over a signature $\Sigma$. Then $\mathbb T$ is equivalent to a coherent theory if and only if for any coherent formula $\{\vec{x}. \phi\}$ over $\Sigma$, for any family $\{\psi_{i}(\vec{x}) \textrm{ | } i\in I\}$ of geometric formulae in the same context, $\phi \vdash_{\vec{x}} \mathbin{\mathop{\textrm{\huge $\vee$}}\limits_{i\in I}} \psi_{i}$ provable in $\mathbb T$ implies $\phi \vdash_{\vec{x}} \mathbin{\mathop{\textrm{\huge $\vee$}}\limits_{i\in I'}} \psi_{i}$ provable in $\mathbb T$ for some finite subset $I'$ of $I$.

This follows similarly to Theorem \[Onetoposmany\_cartcrit\] by using Lemma \[Onetoposmany\_equiv\](iv) and Proposition \[Onetoposmany\_identif\](iv).

Locally connected theories
--------------------------

The following result gives a syntactic characterization of the class of geometric theories classified by a locally connected (resp. connected and locally connected) topos.

\[Onetoposmany\_loccon\] Let $\mathbb T$ be a geometric theory over a signature $\Sigma$. Then $\mathbb T$ is classified by a locally connected topos (resp. connected and locally connected topos) if and only if 