for any geometric formula $\phi(\vec{x})$ over $\Sigma$ there exists a (unique) family $\{\psi_{i}(\vec{x}) \textrm{ | } i\in I\}$ of $\mathbb T$-indecomposable geometric formulae in the same context such that

\(i) for each $i$, $\psi_{i}$ $\mathbb T$-provably implies $\phi$,

\(ii) for any distinct $i, j\in I$, $\psi_{i}\wedge \psi_{j} \vdash_{\vec{x}} \bot$ is provable in $\mathbb T$, and

\(iii) $\phi \vdash_{\vec{x}} \mathbin{\mathop{\textrm{\huge $\vee$}}\limits_{i\in I}} \psi_{i}$ is provable in $\mathbb T$\
(resp. and $\{[].\top\}$ is $\mathbb T$-indecomposable).

Let us suppose that the classifying topos ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$ of $\mathbb T$ is locally connected. Then, by Lemma D3.3.6 [@El2], given a geometric formula $\phi(\vec{x})$ over $\Sigma$, $y^{\textrm{geom}}_{\mathbb T}(\{\vec{x}. \phi\})$ is uniquely expressible as a coproduct of indecomposable objects of ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$. Now, since $y^{\textrm{geom}}_{\mathbb T}({\cal C}_{\mathbb T}^{\textrm{geom}})$ is closed in ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$ under taking subobjects (${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$ being the $\infty$-pretopos completion of ${\cal C}_{\mathbb T}^{\textrm{geom}}$), we can suppose that all the subobjects of $y^{\textrm{geom}}_{\mathbb T}(\{\vec{x}. \phi\})$ in ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$, and in particular the indecomposable objects arising in our coproduct, are of the form $y^{\textrm{geom}}_{\mathbb T}(c)$ for some $c\in {\cal C}_{\mathbb T}^{\textrm{geom}}$. The condition of the criterion 