proposition: Lemma gastabsP (S : {set rT}) (a : aT) : a \in A -> reflect (forall x, (to x a \in S) = (x \in S)) (a \in 'N(S | to)).
proof: Proof. move=> Aa; apply: (iffP idP) => [nSa x|nSa]; first exact: astabs_act. by rewrite !inE Aa; apply/subsetP=> x; rewrite inE nSa. Qed.
imports: HB(structures) mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat seq path) mathcomp(choice fintype bigop finset fingroup morphism) mathcomp(automorphism quotient action gseries)