### A counter-example against a distributivity law for funcoids

Example There exist funcoids $latex {f}&fg=000000$ and $latex {g}&fg=000000$ such that

$latex \displaystyle ( \mathsf{RLD})_{\mathrm{out}} (g \circ f) \neq ( \mathsf{RLD})_{\mathrm{out}} g \circ ( \mathsf{RLD})_{\mathrm{out}} f. &fg=000000$

Proof: Take $latex {f = {( =)} |_{\Omega}}&fg=000000$ and $latex {g = \mho \times^{\mathsf{FCD}} \left\{ \alpha \right\}}&fg=000000$ for some $latex {\alpha \in \mho}&fg=000000$. Then $latex {( \mathsf{RLD})_{\mathrm{out}} f = \emptyset}&fg=000000$ and thus $latex {( \mathsf{RLD})_{\mathrm{out}} g \circ ( \mathsf{RLD})_{\mathrm{out}} f = \emptyset}&fg=000000$. We have $latex {g \circ f = \Omega \times^{\mathsf{FCD}} \left\{ \alpha \right\}}&fg=000000$. Let’s prove $latex {( \mathsf{RLD})_{\mathrm{out}} (\Omega \times^{\mathsf{FCD}} \left\{ \alpha \right\}) = \Omega \times^{\mathsf{RLD}} \left\{ \alpha \right\}}&fg=000000$.

Really:
$latex ( \mathsf{RLD})_{\mathrm{out}} (\Omega \times^{\mathsf{FCD}} \left\{ \alpha \right\}) = \\ \bigcap {\nobreak}^{\mathsf{RLD}} \mathrm{up} (\Omega \times^{\mathsf{FCD}} \left\{ \alpha \right\}) = \\ \bigcap {\nobreak}^{\mathsf{RLD}} \left\{ K \times \left\{ \alpha \right\} \hspace{1em} | \hspace{1em} K \in \mathrm{up} \Omega \right\} = \\ \bigcap {\nobreak}^{\mathfrak{F}} \left\{ K \hspace{1em} | \hspace{1em} K \in \mathrm{up} \Omega \right\} \times^{\mathsf{RLD}} \left\{ \alpha \right\} = \\ \Omega \times^{\mathsf{RLD}} \left\{ \alpha \right\}.&fg=000000$

Thus $latex {( \mathsf{RLD})_{\mathrm{out}} (g \circ f) = \Omega \times^{\mathsf{RLD}} \left\{ \alpha \right\} \neq \emptyset}&fg=000000$. $latex \Box&fg=000000$