Canonical Injection on Group Direct Product is Monomorphism/Proof 1

Proof
From Canonical Injection is Injection we have that the canonical injections are in fact injective.

It remains to prove the morphism property.

Let $x, y \in \left({G_1, \circ_1}\right)$.

Then:

and the morphism property has been demonstrated to hold for $\operatorname{in}_1$.

Thus $\displaystyle \operatorname{in}_1: \left({G_1, \circ_1}\right) \to \left({G_1, \circ_1}\right) \times \left({G_2, \circ_2}\right)$ has been shown to be an injective group homomorphism and therefore a group monomorphism.

The same argument applies to $\operatorname{in}_2$.