Order of External Direct Product

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

Then the order of $\left({S \times T, \circ}\right)$ is $\left|{S}\right| \times \left|{T}\right|$.

Proof
By definition the order of $\left({S \times T, \circ}\right)$ is $\left|{S}\right| \times \left|{T}\right|$ is the cardinality of the underlying set $S \times T$.

The result follows directly from Cardinality of Cartesian Product.