Section Sigma.
Variable f : nat -> R.
Definition sigma (low high:nat) : R :=
sum_f_R0 (fun k:nat => f (low + k)) (high - low).

Theorem sigma_diff_neg :
forall low high k:nat,
(low <= k)%nat ->
(k < high)%nat -> sigma low k - sigma low high = - sigma (S k) high.