$prop:lexprod-indsub$, (2) ⇔ (3). Forwards, take the left factor to be a complete graph on the vertices of $F$: every coefficient $hom(F / 𝓡, K)$ is then positive, since a quotient of $F$ has at most as many vertices as $F$. The formula therefore exhibits a linear combination of the counts from the disjoint unions of the classes that $≡[\mathcal{F}]$ determines, and the lemma on determined linear combinations places each such disjoint union in $cl \mathcal{F}$. Applying this to the partition whose classes are those of $F[U]$ together with a singleton for each vertex outside $U$ gives $F[U]$ with isolated vertices attached; $lem:minors$ strips them off. Backwards, (1) ⇒ (2) applied to $cl \mathcal{F}$.