Corollary: Conditional placement coherence

If the prescriptions (eq:firstprescription)–(eq:mixedprescription) define actual coefficient maps satisfying admission and descent, and both scalar maps use the same allowed $c$, then (eq:assoc1)–(eq:assoc3) hold.

Proof. At order one, both sides sum ${\mathsf{G}}_{12}+{\mathsf{G}}_{13}+{\mathsf{G}}_{23}$. At order two, both sides sum ${\mathsf{G}}_{12,23}+{\mathsf{G}}_{12,13}+{\mathsf{G}}_{13,23}$. At order three, both equal ${\mathcal{N}}_c(f,g,f^{\prime })1$.