Lemma: Overlap gluing

Linear coefficient prescriptions on two subspaces glue to their sum if and only if they agree on the intersection. For three subspaces, each next prescription must agree on its intersection with the sum of the preceding domains. Pairwise agreement alone need not suffice.

Proof. For two domains define the proposed value on $d_1+d_2$ as the sum of the prescribed values. Independence of decomposition is equivalent to agreement on the intersection. For three domains, apply the two-domain statement to the already glued sum and the third domain.