 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[No.] Take $S=T=\{-2,-1\}$. Then
$$
\sup S=\sup T=-1,
\qquad(\sup S)(\sup T)=1,
$$
but $ST=\{1,2,4\}$ and therefore $\sup(ST)=4$.

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