Section Cinq.
Variable H : Ensemble U.
Variable H_inhabited : Inhabited U H.
Variable H_included_in_G : Included U H G.
Variable stability : endo_operation U H star.
Variable H_Finite : Finite U H.
Let h : forall x y : U, In U H x -> In U H y -> In U H (star x y).
Hint Resolve h.
Definition phi (a : U) (n : nat) : U := exp (pos n) a.
Definition psi := phi.

Theorem remaining :
forall a : U,
In U H a ->
exists r : nat, (exists m : nat, phi a r = phi a (S (S (r + m)))).