are provable in $\mathbb T$.

Let $\mathbb T$ be a geometric theory over a signature $\Sigma$ and $\phi(\vec{x})$ a geometric formula-in-context over $\Sigma$. Then

\(i) we say that $\phi(\vec{x})$ is $\mathbb T$-complete if the sequent ($\phi \vdash_{\vec{x}} \bot$) is not provable in $\mathbb T$, and for every geometric formula $\phi$ in the same context either ($\chi \vdash_{\vec{x}} \bot$) or ($\chi \wedge \phi \vdash_{\vec{x}} \bot$) is provable in $\mathbb T$;

\(ii) we say that $\phi(\vec{x})$ is $\mathbb T$-indecomposable if for any family $\{\psi_{i}(\vec{x}) \textrm{ | } i\in I\}$ of geometric formulae in the same context such that for each $i$, $\psi_{i}$ $\mathbb T$-provably implies $\phi$ and for any distinct $i, j\in I$, $\psi_{i}\wedge \psi_{j} \vdash_{\vec{x}} \bot$ is provable in $\mathbb T$, we have that $\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 $i\in I$;

\(iii) we say that $\phi(\vec{x})$ is $\mathbb T$-irreducible if 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 