Projection on Group Direct Product is Epimorphism/Proof 2

Theorem

Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups.

Let $\struct {G, \circ}$ be the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$.

Then:

$\pr_1$ is an epimorphism from $\struct {G, \circ}$ to $\struct {G_1, \circ_1}$

$\pr_2$ is an epimorphism from $\struct {G, \circ}$ to $\struct {G_2, \circ_2}$

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

Proof

A specific instance of Projection is Epimorphism, where the algebraic structures in question are groups.

$\blacksquare$