induction 1; intuition; simpl in *.
inversion H; clear H; subst.
trivial.
inversion H1; clear H1; subst.
f_equal.
eauto.
eauto.
Qed.

Lemma rel_map_permutation : forall (A B : Type) lsa1 lsa2,
Permutation lsa1 lsa2 ->
forall (rel : A -> B -> Prop),
(forall a b1 b2, rel a b1 -> rel a b2 -> b1 = b2) ->
(forall a, exists b, rel a b) ->
forall lsb1 lsb2,
rel_map rel lsa1 lsb1 ->
rel_map rel lsa2 lsb2 ->
Permutation lsb1 lsb2.