Theorem cauchy_finite : forall (An Bn:nat -> R) (N:nat), (0 < N)%nat -> sum_f_R0 An N * sum_f_R0 Bn N = sum_f_R0 (fun k:nat => sum_f_R0 (fun p:nat => An p * Bn (k - p)%nat) k) N + sum_f_R0 (fun k:nat => sum_f_R0 (fun l:nat => An (S (l + k)) * Bn (N - l)%nat) (pred (N - k))) (pred N).