$prop:lexprod-contract$, (2) ⇔ (3). Forwards, take the right factor to be a complete graph on the vertices of $F$: every coefficient $hom(∐ R ∈ 𝓡, F[R], K)$ is then positive, since the disjoint union of the classes is a graph on $V(F)$ and maps into that complete graph. The formula therefore exhibits a linear combination of the counts from the quotients $F / 𝓡$ that $≡[\mathcal{F}]$ determines, and the lemma on determined linear combinations places each quotient in $cl \mathcal{F}$; every contraction is such a quotient. Backwards, (1) ⇒ (2) applied to $cl \mathcal{F}$.