Definition character {G : {set gT}} :=[qualify a phi : 'CF(G) | [forall i, coord (irr G) i phi \in CEnd Coset.Lemma cfun0_char : (0 : 'CF(G)) \is a character.
Proof. by apply/forallP=> i; rewrite lineart=> [|chi xi /forallP-Nchi /forallP-Nxi]; first exact: cfun0_char.
by apply/forallP=> i; rewrite linearD rpredD /=.

(* Specifications: *)
Proof. by rewrite -cfReprReg cfRepr_char. Qed.
Proof. by apply/cfun_inP=> x Gx; rewrite !cf