Proof using.
intros m n h ns f ns' H_in.
apply collate_map2snd_not_in.
intro.
find_apply_lem_hyp in_remove_all_was_in.
intuition.
Qed.

Lemma collate_map2snd_not_in_related :
forall m n h ns (f : A -> A -> list B) ns',
~ In n ns' ->
R h n ->
In n ns ->
NoDup ns ->
collate A_eq_dec h f (map2snd m (filter_rel R_dec h (remove_all A_eq_dec ns' ns))) h n = f h n ++ [m].