Verbality is strongly join-closed

Statement
Suppose $$G$$ is a group and $$H_i, i \in I$$ are verbal subgroups of $$G$$. Then, the join of subgroups $$\langle H_i \rangle$$ is also a verbal subgroup of $$G$$.

Related facts

 * Verbality is transitive
 * Verbality is quotient-transitive
 * Verbality satisfies image condition

Proof
We simply look at the sets of words for each $$H_i$$, and take the set of products of all such words.