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 $\struct {T, \circ_2}$.

Then:
 * $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be closed


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