Section is_RInt_gen.
Context {V : NormedModule R_AbsRing}.
Definition is_RInt_gen (f : R -> V) (Fa Fb : (R -> Prop) -> Prop) (l : V) :=
filterlimi (fun ab => is_RInt f (fst ab) (snd ab)) (filter_prod Fa Fb) (locally l).
Definition ex_RInt_gen (f : R -> V) (Fa Fb : (R -> Prop) -> Prop) :=
exists l : V, is_RInt_gen f Fa Fb l.
Definition is_RInt_gen_at_point f a b l :
is_RInt_gen f (at_point a) (at_point b) l
<-> is_RInt f a b l.
(** * Basic properties of integrals *)

Lemma ex_RInt_gen_ext {Fa Fb : (R -> Prop) -> Prop}
{FFa : Filter Fa} {FFb : Filter Fb} (f g : R -> V) :
filter_prod Fa Fb (fun ab => forall x, Rmin (fst ab) (snd ab) < x < Rmax (fst ab) (snd ab)
-> f x = g x) ->
ex_RInt_gen f Fa Fb -> ex_RInt_gen g Fa Fb.