Lemma: Conditional gluing

If the local extensions and contacts are defined and satisfy (eq:overlap), they determine a unique distribution on their union. This statement does not establish existence of contacts solving (eq:overlap) in ${\mathfrak{C}}_{\mathrm{part}}$.

Proof. A smooth locally finite partition of unity subordinate to the cover defines a distributional pairing by summing the local pairings. The overlap identities make this pairing independent of the partition and agree with each local distribution. Uniqueness follows from localization of every compact test.