Fixpoint removeFirst(A : Set)(eqd : eq_dec A)(ls : list A) a :=
  match ls with
    | nil => nil
    | a' :: ls' =>
      if (eqd a a') then ls' else a' :: (removeFirst eqd ls' a)
  end.

Lemma removeFirst_in_iff : forall (A : Set)(eqd : eq_dec A)(ls : list A) a1 a2, In a1 (removeFirst eqd ls a2) -> In a1 ls.