Inverses in Group Direct Product/Proof 1

Proof
Let $e_G$ be the identity for $\struct {G, \circ_1}$

Let $e_H$ be the identity for $\struct {H, \circ_2}$.

Then:

So the inverse of $\tuple {g, h}$ is $\tuple {g^{-1}, h^{-1} }$.