Intersection of Subsemigroups

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

Let $\struct {T_1, \circ}$ and $\struct {T_2, \circ}$ be subsemigroups of $\struct {S, \circ}$.

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

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

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

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

The other results follow from this and Intersection is Subset.