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.

Lemma orbit_transversalP G S (P := orbit to G @: S)
(X := orbit_transversal G S) :
[acts G, on S | to] ->
[/\ is_transversal X P S, X \subset S,
{in X &, forall x y, (y \in orbit to G x) = (x == y)}
& forall x, x \in S -> exists2 a, a \in G & to x a \in X].