Full invariance is finite direct power-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., fully invariant subgroup) satisfying a subgroup metaproperty (i.e., finite direct power-closed subgroup property)
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Statement

Statement with symbols

Suppose $H$ is a fully invariant subgroup of a group $G$. For any positive integer $n$, consider the external direct product of $G$ with itself $n$, and denote this by $G^{n}$. Let $H^{n}$ be the subgroup comprising those elements where all coordinates are from within $H$. Then, $H^{n}$ is a fully invariant subgroup of $G^{n}$.