External Direct Product Closure

Theorem
Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be algebraic structures.

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

Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be closed.

Then $\struct {S \times T, \circ}$ is also closed.

Proof
Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be closed.

Let $\tuple {s_1, t_1} \in S \times T$ and $\tuple {s_2, t_2} \in S \times T$.

Then:

demonstrating that $\struct {S \times T, \circ}$ is closed.