External Direct Product Closure/Necessary Condition

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 $\struct {S \times T, \circ}$ be a closed algebraic structure.

Then $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ are also closed.

Proof
Let $\struct {S \times T, \circ}$ be closed.

Let $s_1, s_2 \in S$ and $t_1, t_2 \in T$.

Then:

demonstrating that $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ are closed.