Definition cfclass (A : {set gT}) (phi : 'CF(A)) (B : {set gT}) :=
[seq (phi ^ repr Tx)%CF | Tx in rcosets 'I_B[phi] B].
Local Notation "phi ^: G" := (cfclass phi G) : cfun_scope.

(* This is Isaacs, Theorem (6.2) *)
Lemma Clifford_Res_sum_cfclass i j :
H <| G -> j \in irr_constt ('Res[H, G] 'chi_i) ->
'Res[H] 'chi_i =
'['Res[H] 'chi_i, 'chi_j] *: (\sum_(chi <- ('chi_j ^: G)%CF) chi).