Intersection of Subsemigroups

Theorem
Let $\left({S, \circ}\right)$ be a semigroup.

Let $\left({T_1, \circ}\right)$ and $\left({T_2, \circ}\right)$ be subsemigroups of $\left({S, \circ}\right)$.

Then the intersection of $\left({T_1, \circ}\right)$ and $\left({T_2, \circ}\right)$ is itself a subsemigroup of that $\left({S, \circ}\right)$.

If $\left({T, \circ}\right)$ is that intersection of $\left({T_1, \circ}\right)$ and $\left({T_2, \circ}\right)$, it follows that $\left({T, \circ}\right)$ is also a subsemigroup of both $\left({T_1, \circ}\right)$ and $\left({T_2, \circ}\right)$.

Proof
Let $T = T_1 \cap T_2$ where $T_1, T_2$ are subsemigroups of $\left({S, \circ}\right)$. Then:

Thus $\left({T, \circ}\right)$ is closed, and is therefore a semigroup from the Subsemigroup Closure Test.

The other results follow from this and Intersection is Subset.