Section ActsSetop.
Variables (A : {set aT}) (S T : {set rT}).
Hypotheses (AactS : [acts A, on S | to]) (AactT : [acts A, on T | to]).
Definition orbit_transversal A S := transversal (orbit to A @: S) S.
Canonical astab_group S := group (group_set_astab S).
Canonical astabs_group S := group (group_set_astabs S).
End OrbitStabilizer.

Lemma atransPin G S :
G \subset D -> [transitive G, on S | to] ->
forall x, x \in S -> orbit to G x = S.