Canonical Injection is Right Inverse of Projection

Theorem
Let $\left({G_1, \circ_1}\right)$ and $\left({G_2, \circ_2}\right)$ be groups with identity elements $e_1$ and $e_2$ respectively.

Let $\left({G_1 \times G_2, \circ}\right)$ be the group direct product of $\left({G_1, \circ_1}\right)$ and $\left({G_2, \circ_2}\right)$

Let:
 * $\operatorname{pr}_1: \left({G_1 \times G_2, \circ}\right) \to \left({G_1, \circ_1}\right)$ be the first projection from $\left({G_1 \times G_2, \circ}\right)$ to $\left({G_1, \circ_1}\right)$
 * $\operatorname{pr}_2: \left({G_1 \times G_2, \circ}\right) \to \left({G_2, \circ_2}\right)$ be the second projection from $\left({G_1 \times G_2, \circ}\right)$ to $\left({G_2, \circ_2}\right)$.

Let:
 * $\operatorname{in}_1: \left({G_1, \circ_1}\right) \to \left({G_1 \times G_2, \circ}\right)$ be the canonical injection from $\left({G_1, \circ_1}\right)$ to $\left({G_1 \times G_2, \circ}\right)$


 * $\operatorname{in}_2: \left({G_2, \circ_2}\right) \to \left({G_1 \times G_2, \circ}\right)$ be the canonical injection from $\left({G_2, \circ_2}\right)$ to $\left({G_1 \times G_2, \circ}\right)$.

Then:
 * $(1): \quad \operatorname{pr}_1 \circ \operatorname{in}_1 = I_{G_1}$
 * $(2): \quad \operatorname{pr}_2 \circ \operatorname{in}_2 = I_{G_2}$

where $I_{G_1}$ and $I_{G_2}$ are the identity mappings on $G_1$ and $G_2$ respectively.

Proof
This is a specific instance of External Direct Product of Projection with Canonical Injection, where the algebraic structures in question are groups.