Homomorph-containment is finite direct power-closed

Statement
Suppose $$H$$ is a homomorph-containing 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 homomorph-containing subgroup.

Similar facts

 * Full invariance is finite direct power-closed
 * Verbality is finite direct power-closed

Opposite facts

 * Homomorph-containment is not direct power-closed
 * Full invariance is not direct power-closed
 * Verbality is not direct power-closed