Projection is Epimorphism

Theorem
Let $\struct {S, \circ}$ be the external direct product of the algebraic structures $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$.

Then:
 * $\pr_1$ is an epimorphism from $\struct {S, \circ}$ to $\struct {S_1, \circ_1}$
 * $\pr_2$ is an epimorphism from $\struct {S, \circ}$ to $\struct {S_2, \circ_2}$

where $\pr_1$ and $\pr_2$ are the first and second projection respectively of $\struct {S, \circ}$.

Proof
From Projection is Surjection, $\pr_1$ and $\pr_2$ are surjections.

We now need to show they are homomorphisms.

Let $s, t \in \struct {S, \circ}$ where $s = \tuple {s_1, s_2}$ and $t = \tuple {t_1, t_2$.

Then:

Thus the morphism property is demonstrated for both $\pr_1$ and $\pr_2$.