proposition: Lemma pnElemE p n A : prime p -> 'E_p^n(A) = [set E in 'E_p(A) | #|E| == (p ^ n)%N].
proof: Proof. move/pfactorK=> pnK; apply/setP=> E; rewrite 3!inE. case: (@andP (E \subset A)) => //= [[_]] /andP[/p_natP[k ->] _]. by rewrite pnK (can_eq pnK). Qed.
imports: mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path choice) mathcomp(div fintype finfun bigop finset prime binomial) mathcomp(fingroup morphism perm automorphism action) mathcomp(quotient gfunctor gproduct ssralg countalg finalg zmodp) mathcomp(cyclic pgroup gseries nilpotent sylow)