(* The explicit value of the inverse is needed for the proof of the second    *)
(* orthogonality relation.                                                    *)
Let XX'_1: X *m X' = 1%:M.

(* This is Isaacs, Theorem (6.32) (due to Brauer). *)
Lemma card_afix_irr_classes (ito : action A (Iirr G)) (cto : action A _) a :
a \in A -> [acts A, on classes G | cto] ->
(forall i x y, x \in G -> y \in cto (x ^: G) a ->
'chi_i x = 'chi_(ito i a) y) ->
#|'Fix_ito[a]| = #|'Fix_(classes G | cto)[a]|.