proposition: Lemma cfun_complement A : A <| G -> ('CF(G, A) + 'CF(G, G :\: A)%SET = 'CF(G))%VS.
proof: Proof. case/andP=> sAG nAG; rewrite -cfunGid [rhs in _ = rhs]cfun_on_sum. rewrite (bigID (fun B => B \subset A)) /=. congr (_ + _)%VS; rewrite cfun_on_sum; apply: eq_bigl => /= xG. rewrite andbAC; apply/esym/andb_idr=> /andP[/imsetP[x Gx ->] _]. by rewrite class_subG. rewrite -andbA; apply: andb_id2l => /imsetP[x Gx ->]. by rewrite !class_sub_norm ?normsD ?normG // inE andbC. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path) mathcomp(div choice fintype tuple finfun bigop prime order) mathcomp(ssralg poly finset fingroup morphism perm) mathcomp(automorphism quotient finalg action gproduct) mathcomp(zmodp commutator cyclic center pgroup sylow) mathcomp(matrix vector falgebra ssrnum algC algnum) mathcomp(archimedean)