Definition close_tm_rec_erase_tm := fun x => proj1 (close_tm_erase_all x).
Definition close_co_rec_erase_tm := fun x => proj1 (close_co_erase_all x).

Lemma close_tm_erase_tm
: ∀ (x : tmvar) (a : tm), close_tm x (erase a) = erase (close_tm x a).