Fully invariant implies finite direct power-closed characteristic

Statement
Any fully invariant subgroup of a group is a finite direct power-closed characteristic subgroup.

Converse
The converse is not true: finite direct power-closed not implies fully invariant.

Other related facts

 * Bound-word implies finite direct power-closed characteristic
 * Full invariance is finite direct power-closed
 * Characteristicity is not finite direct power-closed

Facts used

 * 1) uses::Fully invariant implies characteristic
 * 2) uses::Full invariance is finite direct power-closed

Proof
The proof follows directly from facts (1) and (2).