Corollary: Conditional consequences of the candidate readings of f and m

If $f\in {C^{\infty }}_c({\mathbb{R}}^4)$ (candidate (d)), then $K_{47}$, ${K_5}^{[a]}$, ${K_5}^{[z]}$, $K_6$, $K_8$ have supports in $\{b\in \mathrm{supp}f\}$. If moreover ${E^{\mathrm{A}}}_m$ is the kernel of Lemma lem:advanced (candidate (f)), then $F={E^{\mathrm{A}}}_m\ast f$ is smooth with $\mathrm{supp}F\subseteq \mathrm{supp}f+J^{-}$, so the hypothesis $F\in C^{\infty }$ of Theorem thm:exist is satisfied, and $K_9$ has $b$-support in $\mathrm{supp}g\cap (\mathrm{supp}f+J^{-})$.

Proof. Each listed kernel is ${\partial }_b$ of a bracket containing the factor $f(b)$; the rest is Lemma lem:advanced.