Infimum of Subset Product in Ordered Group

Theorem
Let $\left({G, \circ, \preceq}\right)$ be an ordered group.

Suppose that subsets $A$ and $B$ of $G$ admit infima in $G$.

Then:
 * $\inf \left({A \circ_{\mathcal P} B}\right) = \inf A \circ \inf B$

where $\circ_{\mathcal P}$ denotes subset product.

Proof
This follows from Supremum of Product and the Duality Principle.

Also see

 * Supremum of Product