Verbality is finite direct power-closed

Statement
Suppose $$H$$ is a verbal subgroup of a group $$G$$. Let $$n$$ be a natural number. Then, in the $$n^{th}$$ direct power $$G^n$$ of $$G$$ (i.e., the external direct product of $$G$$ with itself $$n$$ times) the corresponding subgroup $$H^n$$ is a verbal subgroup. In fact, the same set of words works.

Related facts

 * Full invariance is finite direct power-closed
 * Homomorph-containment is finite direct power-closed

Proof
Given: A group $$G$$, a collection $$C$$ of words, $$H$$ is the subgroup of $$G$$ comprising those elements of $$G$$ that can be expressed as a product of elements that can be expressed in $$G$$ as words from $$C$$. A positive integer $$n$$. An element $$h = (h_1,h_2,\dots,h_n) \in H^n$$.

To prove: $$(h_1,h_2,\dots,h_n)$$ is in the verbal subgroup of $$G^n$$ corresponding to the collection$$C$$.

Proof: We know that there exist words $$w_1,w_2,\dots,w_n$$ such that each $$w_i$$ is expressible as a product of words from $$C$$, and elements $$g_{1,1},\dots,g_{1,m_1},g_{2,1},\dots,g_{2,m_2},\dots,g_{n,1},\dots,g_{n,m_n}$$ such that:

$$h_i = w_i(g_{i,1},\dots,g_{i,m_i})$$

Consider $$w$$ as the word that is the product of the $$w_i$$s, with different input letters for each $$i$$. Then, $$w$$ is also a word generated by $$C$$, and in fact $$h$$ is in the image of the word map corresponding to $$w$$, completing the proof.