Fixpoint sigma_but (l1 : list E) : (E -> Z) -> Z :=
  fun f : E -> Z =>
  match l1 with
  | nil => 0%Z
  | e :: l2 =>
      if eq_E_dec e x0
      then sigma_but l2 f
      else (f e + sigma_but l2 f)%Z
  end.

Lemma sigma_sigma_but : forall l : list E, only_once E eq_E_dec x0 l -> sigma E l f = (sigma_but l f + f x0)%Z.