Definition normal_rcoset_in_lcoset {X : gr} (N : normalsubgr X) : rcoset_in_lcoset N. Proof. induction N as [N normalprop]. simpl. unfold rcoset_in_lcoset. intros g n1. use tpair. - exists ((g^-1) * (pr1 n1) / (g^-1)). use normalprop. - simpl. rewrite (assocax _ (g^-1) _ _). rewrite <- (assocax _ g _ _). rewrite (grrinvax X). rewrite (lunax X). rewrite (grinvinv X). reflexivity. Defined.
Require Import UniMath.MoreFoundations.Subtypes.