Let C'G := algC'G G.
Let closG := @groupC _ G.
Local Notation W i := (@socle_of_Iirr _ G i).
Local Notation "''n_' i" := 'n_(W i).
Local Notation "''R_' i" := 'R_(W i).
Local Notation "''e_' i" := 'e_(W i).

Lemma cfIirrPE J (f : J -> 'CF(G)) (P : pred J) :
(forall j, P j -> f j \in irr G) ->
forall j, P j -> 'chi_(cfIirr (f j)) = f j.