Intersection of Subsemigroups/General Result/Proof 2

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

Proof
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.