Fixpoint SubEquiN (N:nat) (x y:R) (del:posreal) : list R :=
match N with
| O => cons y nil
| S p => cons x (SubEquiN p (x + del) y del)
end.
Definition max_N (a b:R) (del:posreal) (h:a < b) : nat :=
let (N,_) := maxN del h in N.
Definition SubEqui (a b:R) (del:posreal) (h:a < b) : list R :=
SubEquiN (S (max_N del h)) a b del.

Lemma RiemannInt_P12 :
forall (f g:R -> R) (a b l:R) (pr1:Riemann_integrable f a b)
(pr2:Riemann_integrable g a b)
(pr3:Riemann_integrable (fun x:R => f x + l * g x) a b),
a <= b -> RiemannInt pr3 = RiemannInt pr1 + l * RiemannInt pr2.