Supremum of Subset Product in Ordered Group

Theorem
Let $\left({S, \circ, \preceq}\right)$ be a totally ordered structure.

Suppose that subsets $A$ and $B$ of $S$ admit suprema in $S$.

Then the supremum of the product $A \circ_{\mathcal P} B$ exists and is equal to $\sup A \circ \sup B$.

Also see

 * Infimum of Product