Definition sf_SF_val_fun (f : R -> R) (a b : R) (n : nat) : StepFun a b.

(* Specifications: *)
Lemma SF_val_subdiv (f : R -> R) (a b : R) (n : nat) : subdivision (sf_SF_val_fun f a b n) = match (Rle_dec a b) with | left _ => seq2Rlist (unif_part a b n) | right _ => seq2Rlist (unif_part b a n) end.
Lemma SF_val_subdiv_val (f : R -> R) (a b : R) (n : nat) : subdivision_val (sf_SF_val_fun f a b n) = match (Rle_dec a b) with | left _ => seq2Rlist (SF_val_ly f a b n) | right _ => seq2Rlist (SF_val_ly f b a n) end.
Lemma SF_val_fun_rw (f : R -> R) (a b : R) (n : nat) (x : R) (Hx : a <= x <= b) : SF_val_fun f a b n x = match (unif_part_nat a b n x Hx) with | inleft H => f (a + (INR (proj1_sig H) + /2) * (b-a) / (INR n + 1)) | inright _ => f (a + (INR n + /2) * (b-a) / (INR n + 1)) end.
Lemma RInt_val_Reals (f : R -> R) (a b : R) (n : nat) : RInt_val f a b n = RiemannInt_SF (sf_SF_val_fun f a b n).
Lemma ex_Im_fct (f : R -> R) (a b : R) : a <> b -> exists x, (fun y => exists x, y = f x /\ Rmin a b < x < Rmax a b) x.