$thm:complement$, (2) ⇔ (3), the main result. Forwards, the signed sum expressing hom(F, Ḡ) is determined by $≡[\mathcal{F}]$; a coefficient analysis shows that the terms belonging to a single edge deletion, respectively to a single edge contraction, cannot cancel, so $cl \mathcal{F}$ is closed under deleting an edge and under contracting an edge, and these two operations already generate all minors. Backwards, (1) ⇒ (2) applied to $cl \mathcal{F}$.