Derived series members are divisibility-closed in nilpotent group

Statement
In a nilpotent group, all members of the derived series are divisibility-closed subgroups.

Facts used

 * 1) uses::Derived subgroup is divisibility-closed in nilpotent group
 * 2) uses::Nilpotency is subgroup-closed
 * 3) uses::Divisibility-closedness is transitive

Proof
The proof follows by combining Facts (1)-(3).