Potentially verbal implies normal

Statement
Suppose $$G$$ is a group and $$H$$ is a potentially verbal subgroup of $$G$$, i.e., there exists a group $$K$$ containing $$G$$ such that $$H$$ is a verbal subgroup of $$K$$.

Then, $$H$$ is a normal subgroup of $$G$$.

Facts used

 * 1) uses::Verbal implies normal
 * 2) uses::Normality satisfies intermediate subgroup condition

Proof
Given: A group $$G$$, a subgroup $$H$$ of $$G$$, a group $$K$$ containing $$G$$ such that $$H$$ is a verbal subgroup of $$K$$.

To prove: $$H$$ is normal in $$G$$.

Proof:


 * 1) By fact (1), $$H$$ is normal in $$K$$.
 * 2) By fact (2), since $$H$$ is normal in $$K$$, and $$G$$ is an intermediate subgroup of $$K$$, then $$H$$ is normal in $$G$$.