proposition: Lemma sum_Iirr_kerD_square H M : H <| K -> M <| K -> M \subset H -> \sum_(i in Iirr_kerD H M) 'chi_i 1%g ^+ 2 = #|K : H|%:R * (#|H : M|%:R - 1).
proof: Proof. move=> nsHK nsMK sMH; have [sHK _] := andP nsHK. rewrite mulrBr mulr1 -natrM Lagrange_index // -!sum_Iirr_ker_square //. apply/esym/(canLR (addrK _)); rewrite /= addrC (big_setID (Iirr_ker H)). by rewrite (setIidPr _) ?Iirr_kerS //. Qed.
imports: mathcomp.ssreflect.ssreflect mathcomp(ssrbool ssrfun eqtype ssrnat seq path div choice fintype) mathcomp(tuple finfun bigop order prime ssralg poly finset center) mathcomp(fingroup morphism perm automorphism quotient action zmodp) mathcomp(gfunctor gproduct cyclic pgroup frobenius) mathcomp(matrix mxalgebra mxrepresentation vector ssrint) mathcomp(ssrnum algC classfun character inertia vcharacter) odd_order(PFsection1 PFsection2 PFsection3 PFsection4)