External Direct Product Closure

Theorem
Let $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$ be algebraic structures.

Let $\left({S \times T, \circ}\right)$ be the external direct product of $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$.

If $\left({S \times T, \circ}\right)$ is closed, then $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$ are also closed.

Proof
Let $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$ be closed.

Let $\left({s_1, t_1}\right) \in S \times T$ and $\left({s_2, t_2}\right) \in S \times T$.

Then:

demonstrating that $\left({S \times T, \circ}\right)$ is closed.