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 intermediately local powering-invariant subgroups of $G$.

Facts used

1. Upper central series members are local powering-invariant in nilpotent group
2. 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.