proposition: Lemma index_quotient_eq : G :&: H \subset K -> K \subset G -> G \subset 'N(H) -> #|G / H : K / H| = #|G : K|.
proof: Proof. move=> sGH_K sKG sGN; rewrite -index_quotient_ker {sKG sGN}//. by rewrite -(indexgI _ K) (setIidPl sGH_K) indexgg muln1. Qed.
imports: HB(structures) mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat seq div) mathcomp(choice fintype prime finset fingroup morphism) mathcomp(automorphism)