Let $E$ be a finite set and let $\mathcal C$ be a collection of nonempty subsets of $E$ such that no member of $\mathcal C$ properly contains another member of $\mathcal C$, and such that circuit elimination holds: whenever $C_1,C_2\in\mathcal C$ are distinct and $e\in C_1\cap C_2$, there is $C_3\in\mathcal C$ with