ite !inE groupM ?(@astab_dom _ _ _ to S) //; apply/subsetP=> x Sx.
by rewrite inE actMin ?(@astab_dom _ _ _ to S) ?(astab_act _ Sx).
Qed.ite !inE groupM ?(@astabs_dom _ _ _ to S) //; apply/subsetP=> x Sx.
by rewrite inE actMin ?(@astabs_dom _ _ _ to S) ?astabs_act.
Qed.

Lemma actK : right_loop invg to.