=> Gx; suffices: x \in rker (irr_repr 1) by case/rkerP.
apply: subsetP x Gx; rewrite rker_submod rfix_mx_rstabC // -irr1_rfix.
by apply: component_mx_id; apply: socle_simp

(* Specifications: *)
& forall x, x \in G -> rG1 x *m B = B *m rG2 x.
Proof. by case. Qed.
Qed.
Qed. Lemma mx_rsim_trans n1 n2 n3
Qed.
Qed. Lemma mx_rsim_abs_irr n1 n2 (rG1 : reprG n1) (rG2 : reprG n2) :
Qed.
Qed.