(** ** SF_sup_real is a StepFun *)

Lemma ad_SF_sup_r (f : R -> R) (a b : R) (n : nat) :
((a <= b) -> adapted_couple (fun x => real (SF_sup_fun f a b n x)) a b
(seq2Rlist (unif_part a b n))
(seq2Rlist (behead (pairmap (fun x y => real (Sup_fct f x y)) 0 (unif_part a b n)))))
/\ (~(a <= b) -> adapted_couple (fun x => real (SF_sup_fun f a b n x)) a b
(seq2Rlist (unif_part b a n))
(seq2Rlist (behead (pairmap (fun x y => real (Sup_fct f x y)) 0 (unif_part b a n))))).