induction ls; intuition; simpl in *.
destruct i; intuition.
destruct i; subst.
omega.
eapply lt_n_S.
eauto.
Qed.

Lemma map_eq_nth_h :
forall (A B C : Set) (f1 : A -> C)(f2 : B -> C)(lsa : list A)(lsb : list B),
length lsa = length lsb ->
map f1 lsa = map f2 lsb ->
forall i def1 def2,
i < length lsa ->
f1 (nth i lsa def1) = f2 (nth i lsb def2).