Complete divisibility-closedness is transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., completely divisibility-closed subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Statement

Suppose $G$ is a group and $H,K$ are subgroups with $H\le K\le G$. Suppose $K$ is a completely divisibility-closed subgroup of $G$ and $H$ is a completely divisibility-closed subgroup of $K$. Then, $H$ is a completely divisibility-closed subgroup of $G$.

Related facts

• Powering-invariance is transitive
• Divisibility-closedness is transitive