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 Heine_cor1 :
forall (f:R -> R) (a b:R),
a < b ->
(forall x:R, a <= x <= b -> continuity_pt f x) ->
forall eps:posreal,
{ delta:posreal |
delta <= b - a /\
(forall x y:R,
a <= x <= b ->
a <= y <= b -> Rabs (x - y) < delta -> Rabs (f x - f y) < eps) }.