Internal Group Direct Product Commutativity/Proof 1

Proof
Let $G$ be the internal group direct product of $H$ and $K$.

Then by definition the mapping:
 * $\phi: H \times K \to G: \map \phi {h, k} = h \circ k$

is a (group) isomorphism from the (external) direct product $\struct {H, \circ \restriction_H} \times \struct {K, \circ \restriction_K}$ onto $\struct {G, \circ}$.

Let the symbol $\circ$ also be used for the operation induced on $H \times K$ by $\circ \restriction_H$ and $\circ \restriction_K$.

Let $h \in H, k \in K$.

Then: