Definition SF_sup_fun (f : R -> R) (a b : R) (n : nat) (x : R) : Rbar :=
match (Rle_dec a b) with
| left _ => SF_fun (SF_sup_seq f a b n) (Finite 0) x
| right _ => SF_fun (SF_sup_seq f b a n) (Finite 0) x
end.
Definition SF_inf_fun (f : R -> R) (a b : R) (n : nat) (x : R) : Rbar :=
match (Rle_dec a b) with
| left _ => SF_fun (SF_inf_seq f a b n) (Finite 0) x
| right _ => SF_fun (SF_inf_seq f b a n) (Finite 0) x
end.

Lemma SF_sup_fun_bound (f : R -> R) (a b : R) (n : nat) (x : R) :
SF_sup_fun f a b n x = SF_sup_fun f b a n x.