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 is_RInt_gen_Chasles {Fa Fc : (R -> Prop) -> Prop}
{FFa : Filter Fa} {FFc : Filter Fc}
(f : R -> V) (b : R) (l1 l2 : V) :
is_RInt_gen f Fa (at_point b) l1 -> is_RInt_gen f (at_point b) Fc l2
-> is_RInt_gen f Fa Fc (plus l1 l2).