Semigroup is Subsemigroup of Itself

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

Then $\struct {S, \circ}$ is a subsemigroup of itself.

Proof
For all sets $S$, $S \subseteq S$, that is, $S$ is a subset of itself.

Thus $\struct {S, \circ}$ is a semigroup which is a subset of $\struct {S, \circ}$, and therefore a subsemigroup of $\struct {S, \circ}$.