= Incidence equivalence in a set family
{title2=$x\sim y\iff\forall B\in\mathcal F:\ (x\in B\iff y\in B)$}
Two ground elements are incidence-equivalent when they belong to exactly the same members of a <set family>. Their incidence columns are identical. For the <set basis problem>, deleting one of two such elements preserves the answer. A basis before deletion can be restricted to the smaller ground <set>. Conversely, if $x$ remains and $y$ was deleted, lift each reduced basis <set> $A$ to $A\cup\{y\}$ when $x\in A$, and leave it unchanged otherwise. In every <union> of lifted basis <sets>, $y$ appears exactly when $x$ appears, which restores precisely the original input <sets>. Repeated deletion thus leaves one representative of every incidence class.
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