Upper central series members are intermediately local powering-invariant in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group. Then, the members of the upper central series of $$G$$ (such as the  center and  second center) are all  proves property satisfaction of::intermediately local powering-invariant subgroups of $$G$$.

Facts used

 * 1) uses::Upper central series members are local powering-invariant in nilpotent group
 * 2) uses::Local powering-invariant subgroup containing the center is intermediately local powering-invariant in nilpotent group

Proof
Note that the trivial subgroup is obviously local powering-invariant, and hence intermediately local powering-invariant. For all the later members of the upper central series, Facts (1) and (2) together prove that they are all intermediately local powering-invariant.