Compact support continuous primitive representation

Use F3 in $V$ to choose $\theta \in \mathcal{D}(V)$ equal to one near $K$, and put $L=\mathrm{supp}\theta$. By F1 and F5 there is continuous $f$ on $\mathrm{\Omega }$ with $u={\mathrm{\partial }}^{\alpha }{u}_f$ on ${\mathcal{D}}_L$, where each ${\alpha }_i=m+2$. For every test $\phi$, F4 gives $u(\phi )=u(\theta \phi )$, and $\theta \phi \in {\mathcal{D}}_L$, hence $u=\theta {\mathrm{\partial }}^{\alpha }{u}_f$ globally on $\mathrm{\Omega }$.

For any distribution $w$, F2 implies the inverse product identity [step 1.1, F2, algebra] $\theta {\mathrm{\partial }}^{\alpha }w={\sum }_{\beta \le \alpha }(-1{)}^{\mathrm{\mid }\alpha -\beta \mathrm{\mid }}\left(\genfrac{}{}{0px}{}{\alpha }{\beta }\right){\mathrm{\partial }}^{\beta }(({\mathrm{\partial }}^{\alpha -\beta }\theta )w)\mathrm{.}$ Indeed expand each derivative on the right by F2. The coefficient of $({\mathrm{\partial }}^{\alpha -\gamma }\theta ){\mathrm{\partial }}^{\gamma }w$ is $\left(\genfrac{}{}{0px}{}{\alpha }{\gamma }\right){\sum }_{\delta \le \alpha -\gamma }(-1{)}^{\mathrm{\mid }\alpha -\gamma -\delta \mathrm{\mid }}\left(\genfrac{}{}{0px}{}{\alpha -\gamma }{\delta }\right)$. The sum is the product of the binomial expansions of $(1-1{)}^{{\alpha }_i-{\gamma }_i}$, so it is zero unless $\gamma =\alpha$, when it is one. Thus precisely the left side remains.

Apply step 2.1 to $w=u_f$ and define $g_{\beta }=(-1{)}^{\mathrm{\mid }\alpha -\beta \mathrm{\mid }}\left(\genfrac{}{}{0px}{}{\alpha }{\beta }\right)({\mathrm{\partial }}^{\alpha -\beta }\theta )f$. These functions are continuous and supported in $L\subseteq V$. Multiplying a regular functional by the smooth factor yields its pointwise product by the defining integral, so step 1.1 and the identity give the required sum. There are only ${\prod }_i(m+3)$ indices. If $K$ is empty, $u=0$ by support locality and all functions may be zero. For $m=0$ the bound is still ${\beta }_i\le 2$. No further infinite selection occurs. $\mathrm{\square }$