External Direct Product Identity/General Result

Theorem
Let $\displaystyle \left({S, \circ}\right) = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$.

Let $e_1, e_2, \ldots, e_n$ be the identity elements of $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$ respectively.

Then $\left({e_1, e_2, \ldots, e_n}\right)$ is the identity element of $\left({S, \circ}\right)$.

Proof
Let $s := \left({s_1, s_2, \ldots, s_n}\right)$ be an arbitrary element of $\left({S_1, \circ_1}\right) \times \left({S_2, \circ_2}\right) \times \cdots \times \left({S_n, \circ_n}\right)$.

Let $e := \left({e_1, e_2, \ldots, e_n}\right)$.

Then:

and: