Power Structure of Semigroup Ordered by Supersets is Ordered Semigroup

Theorem
Let $\struct {S, \circ}$ be a semigroup.

Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.

Let $\struct {\powerset S, \circ_\PP, \supseteq}$ be the ordered structure formed from $\struct {\powerset S, \circ_\PP}$ and the superset relation.

Then $\struct {\powerset S, \circ_\PP, \supseteq}$ is an ordered semigroup.

Proof
From Power Structure of Semigroup Ordered by Subsets is Ordered Semigroup, $\struct {\powerset S, \circ_\PP, \subseteq}$ is an ordered semigroup.

The result then follows from Dual of Ordered Semigroup is Ordered Semigroup.