intros i j work ds ni1 ni2 ai n eqv12 lt1.
apply lt_S_n.
apply le_lt_trans with (nweight_Sequent work ((i, j) :: ds) ni1 ai);
try assumption; clear lt1.
unfold nweight_Sequent in |- *.
generalize (nweight_ai ai); clear ai; intros ai.
rewrite (nweight_eqv_ni ni1 ni2 eqv12); clear eqv12.
generalize (nweight_ni ni2 + ai); clear ni2 ai n; intros n.
simpl in |- *.
generalize (nweight_work work); clear work; intros W.
generalize (nweight_ds ds); clear ds; intros ds.
rewrite <- (plus_Snm_nSm W (ds + n)).
simpl in |- *; trivial.
Qed.

Lemma nweight_sequent_left_disj_right :
forall (i j : Int) (work : nf_list) (ds : disjs) (ni1 ni2 : nested_imps)
(ai : atomic_imps) (n : nat),
eqv_ni ni1 ni2 ->
nweight_Sequent work ((i, j) :: ds) ni1 ai < S n ->
nweight_Sequent (NAtom j :: work) ds ni2 ai < n.