-(*
-lemma apply_inj_aux: ∀f1,f2,j1,j2,i1,i2. f1@❴i1❵ = j1 → f2@❴i2❵ = j2 →
- j1 = j2 → f1 ≗ f2 → i1 = i2.
-/2 width=6 by at_inj/ qed-.
-*)
+
+theorem apply_inj: ∀f,i1,i2,j. f@❨i1❩ = j → f@❨i2❩ = j → i1 = i2.
+/2 width=4 by at_inj/ qed-.
+
+corec theorem nstream_eq_inv_ext: ∀f1,f2. (∀i. f1@❨i❩ = f2@❨i❩) → f1 ≗ f2.
+* #n1 #f1 * #n2 #f2 #Hf @eq_seq
+[ @(Hf 0)
+| @nstream_eq_inv_ext -nstream_eq_inv_ext #i
+ lapply (Hf 0) >apply_O1 >apply_O1 #H destruct
+ lapply (Hf (↑i)) >apply_S1 >apply_S1 #H
+ /3 width=2 by injective_plus_r, injective_S/
+]
+qed-.