Verbality is transitive

Statement
Suppose $$H \le K \le G$$ are groups with each verbal in the next (i.e., $$G$$ is a group, $$K$$ is a verbal subgroup of $$G$$, and $$H$$ is a verbal subgroup of $$K$$). Then, $$H$$ is a verbal subgroup of $$G$$.

Related facts

 * Verbality is quotient-transitive
 * Verbality is strongly join-closed
 * Verbality satisfies image condition

Proof idea
The idea is to "compose" the words by substituting. Explicitly, any element of $$H$$ can be written as a word of a certain type in terms of elements of $$K$$, and each of those elements of $$K$$ can be written as words of certain types in the element of $$G$$. We plug in those word expressions. Explicitly, if:

$$h = w(k_1,k_2,\dots,k_n)$$

where:

$$k_i = w_i(\mbox{elements of } G)$$

then:

$$h = (w \circ (w_1,w_2,\dots,w_n))(\mbox{elements of } G)$$