(***)
intros. red. intros g1 g2 H. elim(classic(Pred_fun(Stab2_s G S
f LA1 LA2 p)g1)). intro H1. assert(H2:Pred_fun(Stab2_s G S f
LA1 LA2 p)g2). simpl;simpl in H1. apply Trans with(f(couple g1
p)). intuition. auto. apply Trans with(Build_subtype H1).
unfold G2Stab;unfold sumbool_rect;destruct
excluded_middle_informative. simpl;red;apply Refl. intuition.
unfold G2Stab;unfold sumbool_rect;destruct
excluded_middle_informative. intuition. intuition.
(***)
intro H1. assert(H2:~Pred_fun(Stab2_s G S f LA1 LA2 p)g2).
unfold not;intro H2. apply H1. simpl;simpl in H2. apply Trans
with(f(couple g2 p)). intuition. auto. assert(H0:Pred_fun
(Stab2_s G S f LA1 LA2 p)(zero G)). simpl;apply LA1. apply
Trans with(Build_subtype H0). unfold G2Stab;unfold
sumbool_rect;destruct excluded_middle_informative. intuition.
simpl;red;apply Refl. unfold G2Stab;unfold sumbool_rect;
destruct excluded_middle_informative. intuition. simpl;red;
apply Refl. Qed.

(* Specifications: *)
Lemma orbstab2:forall(G:group)(S:Setoid)(f:Map(cart G S)S)
(LA1:forall(s:S),f(couple(zero G)s)='s)
(LA2:forall(g1 g2:G)(s:S),
f(couple(g1+'g2)s)='f(couple g1(f(couple g2 s))))
(p:S)(n:Nat)(H:has_n_elements n G),
exists c:Nat,exists k:Nat,
has_n_elements c(Orb7_s G S f LA1 LA2 p)/\
has_n_elements k(Stab2_s G S f LA1 LA2 p)/\n='c*k.