proposition: Lemma cfInd_sum_Inertia : [/\ {in calA, forall t, 'Ind 'chi_t \in irr G}, {in calA, forall t, 'chi_(AtoB t) = 'Ind 'chi_t}, {in calA &, injective AtoB}, AtoB @: calA =i calB & 'Ind[G] theta = \sum_(t in calA) e_ t *: 'Ind 'chi_t].
proof: Proof. have [AtoBirr AtoBinj defB _ _] := constt_Inertia_bijection s nsHG. split=> // [i Ai|]; first exact/cfIirrE/AtoBirr. rewrite -(cfIndInd _ sTG sHT) {1}['Ind theta]cfun_sum_constt linear_sum. by apply: eq_bigr => i _; rewrite linearZ. Qed.
imports: mathcomp.ssreflect.ssreflect mathcomp(ssrbool ssrfun eqtype ssrnat seq path div choice fintype) mathcomp(tuple finfun bigop prime order ssralg finset fingroup morphism) mathcomp(perm automorphism quotient action zmodp finalg center commutator) mathcomp(poly cyclic pgroup nilpotent matrix mxalgebra mxrepresentation) mathcomp(vector falgebra fieldext ssrnum algC rat algnum galois) mathcomp(classfun character inertia integral_char vcharacter)