Retract implies local divisibility-invariant

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., retract) must also satisfy the second subgroup property (i.e., local divisibility-invariant subgroup)
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Statement

Suppose $G$ is a group and $H$ is a retract of $G$. Then, $H$ is a local divisibility-invariant subgroup of $G$. In other words: suppose $n$ is a natural number. Suppose $h\in H$ is such that there exists $x\in G$ such that $x^{n}=h$. Then, there exists $y\in H$ (possibly equal to $x$ such that $y^{n}=h$.