Verbal subgroup of abelian group implies divisibility-closed

Statement
Suppose $$G$$ is an abelian group and $$H$$ is a verbal subgroup of $$G$$ (so $$H$$ is a verbal subgroup of abelian group). Then, $$H$$ is a divisibility-closed subgroup of $$G$$. In other words, for any prime number $$p$$ such that $$G$$ is $$p$$-divisible, $$H$$ is also $$p$$-divisible.

Facts used

 * 1) uses::Verbal subgroup equals power subgroup in abelian group

Proof
The proof follows from Fact (1) and a little more work.