Definition SF_cut_down' (sf : @SF_seq R) (x : R) x0 :=
  let s := seq_cut_down' ((SF_h sf,x0) :: (SF_t sf)) x x0 in
  mkSF_seq (fst (head (SF_h sf,x0) s)) (behead s).Definition SF_cut_up' (sf : @SF_seq R) (x : R) x0 :=
  let s := seq_cut_up' ((SF_h sf,x0) :: (SF_t sf)) x x0 in
  mkSF_seq (fst (head (SF_h sf,x0) s)) (behead s).

Lemma SF_Chasles {V : ModuleSpace R_AbsRing} (f : R -> V) (s : SF_seq) x x0 : (SF_h s <= x <= last (SF_h s) (unzip1 (SF_t s))) -> Riemann_sum f s = plus (Riemann_sum f (SF_cut_down' s x x0)) (Riemann_sum f (SF_cut_up' s x x0)).