External Direct Product of Groups is Group

Theorem
Let $\left({G_1, \circ_1}\right)$ and $\left({G_2, \circ_2}\right)$ be groups.

Let $\left({G_1 \times G_2, \circ}\right)$ be the external direct product of $G_1$ and $G_2$.

Then $\left({G_1 \times G_2, \circ}\right)$ is a group

Proof
Follows directly from:
 * External Direct Product Associativity
 * External Direct Product Commutativity
 * External Direct Product Identity
 * External Direct Product Inverses

Closure is trivial.