Inverses in Group Direct Product

Theorem
Let $\struct {G \times H, \circ}$ be the group direct product of the two groups $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$.

Let $g^{-1}$ be an inverse of $g \in \struct {G, \circ_1}$.

Let $h^{-1}$ be an inverse of $h \in \struct {H, \circ_2}$.

Then $\tuple {g^{-1}, h^{-1} }$ is the inverse of $\tuple {g, h} \in \struct {G \times H, \circ}$.