End NilpotentProps.
Section LowerCentral.
Variable gT : finGroupType.
Implicit Types (A B : {set gT}) (G H : {group gT}).

Lemma der_bigcprod n I r P (F : I -> {set gT}) G :
\big[cprod/1]_(i <- r | P i) F i = G ->
\big[cprod/1]_(i <- r | P i) (F i)^`(n) = G^`(n).