Direct factor is transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., direct factor) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Statement

Suppose $G$ is a group and $H\leq K\leq G$. Suppose that $H$ is a direct factor of $K$ and $K$ is a direct factor of $G$. Then, $H$ is a direct factor of $G$.

Related facts

• Direct product is associative up to natural isomorphism
• Direct product is commutative up to natural isomorphism