Variables (A B C D : {set gT}).
Hypotheses (nBA : A \subset 'N(B)) (nCA : A \subset 'N(C)).

(* An elementary proof that subgroups of index 2 are normal; it is almost as  *)
(* short as the "advanced" proof using group actions; besides, the fact that  *)
(* the coset is equal to the complement is used in extremal.v.                *)
Lemma rcoset_index2 G H x :
H \subset G -> #|G : H| = 2 -> x \in G :\: H -> H :* x = G :\: H.