External Direct Product Identity/General Result

Theorem
Let $\ds \struct {\SS, \circ} = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$.

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

Then $\tuple {e_1, e_2, \ldots, e_n}$ is the identity element of $\struct {\SS, \circ}$.

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

Let $e := \tuple {e_1, e_2, \ldots, e_n}$.

Then:

and: