(* Lemma RInt_comp_opp (f : R -> V) (a b : R) : ex_RInt f (-a) (-b) -> RInt (fun y => f (- y)) a b = opp (RInt f (-a) (-b)). *)

(* Specifications: *)
Lemma RInt_comp_lin (f : R -> V) (u v a b : R) :
ex_RInt f (u * a + v) (u * b + v) ->
RInt (fun y => scal u (f (u * y + v))) a b = RInt f (u * a + v) (u * b + v).
Lemma RInt_Chasles :
forall f a b c,
ex_RInt f a b -> ex_RInt f b c ->
plus (RInt f a b) (RInt f b c) = RInt f a c.
Lemma RInt_scal (f : R -> V) (a b l : R) :
ex_RInt f a b ->
RInt (fun x => scal l (f x)) a b = scal l (RInt f a b).
Lemma RInt_opp (f : R -> V) (a b : R) :
ex_RInt f a b ->
RInt (fun x => opp (f x)) a b = opp (RInt f a b).
Lemma RInt_plus :
forall f g a b, ex_RInt f a b -> ex_RInt g a b ->
RInt (fun x => plus (f x) (g x)) a b = plus (RInt f a b) (RInt g a b).
Lemma RInt_minus :
forall f g a b, ex_RInt f a b -> ex_RInt g a b ->
RInt (fun x => minus (f x) (g x)) a b = minus (RInt f a b) (RInt g a b).