proposition: Lemma nElemI n G H : 'E^n(G :&: H) = 'E^n(G) :&: subgroups H.
proof: Proof. apply/setP=> E; apply/nElemP/setIP=> [[p] | []]. by rewrite pnElemI; case/setIP; split=> //; apply/nElemP; exists p. by case/nElemP=> p EpnG_E sHE; exists p; rewrite pnElemI inE EpnG_E. 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)