External Direct Product Identity

Theorem
Let $$\left({S \times T, \circ}\right)$$ be the external direct product of the two algebraic structures $$\left({S, \circ_1}\right)$$ and $$\left({T, \circ_2}\right)$$.

If:
 * $$e_S$$ is the identity for $$\left({S, \circ_1}\right)$$, and:
 * $$e_T$$ is the identity for $$\left({T, \circ_2}\right)$$;

then $$\left({e_S, e_T}\right)$$ is the identity for $$\left({S \times T, \circ}\right)$$.

Proof
So the identity is $$\left({e_S, e_T}\right)$$.