Existence of Unique Subsemigroup Generated by Subset/Proof 2

Proof
Let $\mathbb S$ be the set of all subsemigroups of $S$.

From Set of Subsemigroups forms Complete Lattice:
 * $\struct {\mathbb S, \subseteq}$ is a complete lattice.

where for every set $\mathbb H$ of subsemigroups of $S$:
 * the infimum of $\mathbb H$ necessarily admitted by $\mathbb H$ is $\ds \bigcap \mathbb H$.

Hence the result, by definition of infimum.