Complete divisibility-closedness is transitive

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