induction s.
simpl in |- *.
tauto.
simpl in |- *.
intros x z.
elim (eq_dart_dec d x).
intro.
apply IHs.
simpl in |- *.
intros.
elim H.
intro.
rewrite <- H0.
intro.
rewrite H1 in b.
tauto.
apply IHs.
Qed.

Lemma exds_Ds_exds:forall(s:set)(x z:dart),
exds (Ds s x) z -> exds s z.