(**)
intros U F F' F'' F''' u u'''' A B A' B' H. generalize B'. rewrite<-H. apply
u''''. Qed.

(*----------------------------------------------------------------------------- The Lebesgue integral of an indicator function is just the measure of the event interval. -----------------------------------------------------------------------------*)
Lemma lebint_P1:
forall(U:Type)(F:Ensemble(Ensemble U))(F':sigalg_0 _ F)(F'':sigalg_1 _ F)(F''':
sigalg_2 _ F)(u:forall A,F A->R)(u':msur_0 _ _ F' F'' F''' u)(u'':msur_1 _ _ F'
F'' F''' u)(u''':msur_2 _ _ F' F'' F''' u)(u'''':msur_3 _ _ F' F'' F''' u)(f:U
->R)(f':forall x:U,(f x>=0)%R)(f'':msurable _ _ F' F'' F''' f)(E:Ensemble U)(E'
:F E),
forall(A:Ensemble U)(A':F A)(A'':Included _ A E),
lebint_s _ _ F' F'' F''' u u' u'' u''' u'''' 2(fun i:nat=>INR i)(fun i:nat=>if
eq_nat_dec i 0 then Complement _ A else if eq_nat_dec i 1 then A else Empty_set
_)(simple_P4 _ _)(simple_P5 _ _)(simple_P6 _ _)(simple_P7 _ _)(simple_P8 _ _ F'
F'' F''' A A')E E'=u A A'.