Inverses in Group Direct Product/Proof 2
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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 $\tuple {g, h} \in \struct {G \times H, \circ}$.
Proof
A specific instance of External Direct Product Inverses, where the algebraic structures in question are groups.
$\blacksquare$