Section RInt_gen.
Context {V : CompleteNormedModule R_AbsRing}.
Definition RInt_gen (f : R -> V) (a b : (R -> Prop) -> Prop) :=
iota (is_RInt_gen f a b).
End RInt_gen.

Lemma is_RInt_gen_Derive {Fa Fb : (R -> Prop) -> Prop} {FFa : Filter Fa} {FFb : Filter Fb}
(f : R -> R) (la lb : R) :
filter_prod Fa Fb
(fun ab => forall x : R, Rmin (fst ab) (snd ab) <= x <= Rmax (fst ab) (snd ab) -> ex_derive f x)
-> filter_prod Fa Fb
(fun ab => forall x : R, Rmin (fst ab) (snd ab) <= x <= Rmax (fst ab) (snd ab) -> continuous (Derive f) x)
-> filterlim f Fa (locally la) -> filterlim f Fb (locally lb)
-> is_RInt_gen (Derive f) Fa Fb (lb - la).