generalize (join_ex_identities s); intros [s0 [A [s' B]]].
exists s0; split... assert (s = s'). apply (A _ _ B). subst s...
Qed.

Lemma lseg_nilnil sigma s :
space_denote sigma s =
(space_atom_denote (Lseg Nil Nil) * space_denote sigma)%pred s.