Full invariance is finite direct power-closed

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 $$G^n$$.

Opposite facts

 * Full invariance is not direct power-closed
 * Characteristicity is not finite direct power-closed

Similar facts

 * Homomorph-containment is finite direct power-closed
 * Normality-preserving endomorphism-invariance is finite direct power-closed
 * Verbality is finite direct power-closed