$\mathbb T$ be a geometric theory over a signature $\Sigma$. Then $\mathbb T$ is equivalent to a cartesian theory if and only if for any cartesian formula $\{\vec{x}. \phi\}$ over $\Sigma$, for any family $\{\theta_{i} \textrm{ | } i\in I\}$ of $\mathbb T$-provably functional geometric formulae $\{\vec{x_{i}}, \vec{x}.\theta_{i}\}$ from $\{\vec{x_{i}}. \phi_{i}\}$ to $\{\vec{x}. \phi\}$ such that $\phi \vdash_{\vec{x}} \mathbin{\mathop{\textrm{\huge $\vee$}}\limits_{i\in I}}(\exists \vec{x_{i}})\theta_{i}$ is provable in $\mathbb T$, there exist $i\in I$ and a $\mathbb T$-provably functional geometric formula $\{\vec{x}, \vec{x_{i}}. \theta'\}$ from $\{\vec{x}. \phi\}$ to $\{\vec{x_{i}}. \phi_{i}\}$ such that $\phi \vdash_{\vec{x}} (\exists \vec{x_{i}})(\theta' \wedge \theta_{i})$ is provable in $\mathbb T$.

Let us suppose that $\mathbb T$ is cartesian. Then the property of $\{\vec{x}. \phi\}$ in the statement of the proposition is equivalent, by Lemma \[Onetoposmany\_equiv\](iii), to saying that $y^{\textrm{geom}}_{\mathbb T}(\{\vec{x}. \phi\})$ is irreducible in ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$. But this condition corresponds, under the equivalence $[{{\cal C}_{\mathbb T}^{\textrm{cart}}}^{\textrm{op}}, {{\bf Set }}]\simeq {{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$, to saying that $y^{\textrm{cart}}_{\mathbb T}(\{\vec{x}. \phi\})$ is irreducible (equivalently, indecomposable projective) in $[{{\cal C}_{\mathbb T}^{\textrm{cart}}}^{\textrm{op}}, {{\bf Set }}]$, and this true (cfr. Proposition \[Onetoposmany\_identif\](iii)).

Conversely, if $\mathbb T$ is geometric and for any cartesian formula $\phi(\vec{x})$ over $\Sigma$, $y^{\textrm{geom}}_{\mathbb T}(\{\vec{x}. \phi\})$ 