(**)
intros. unfold lebint_s. simpl. assert(H:Intersection _(Empty_set U)E=Empty_set
U). apply Extensionality_Ensembles. split. red. intros x H1. elim H1. auto. red
. intros. split. auto. red in H. intuition.
(**)
replace(u(Intersection _(Empty_set _)E)(sigalg_P2 U F F' F'' F'''(Empty_set U)E
(simple_P1 U F F' F'' F''' 2(fun i=>INR i)(fun i=>if eq_nat_dec i 0 then
Complement _ A else if eq_nat_dec i 1 then A else Empty_set _)(simple_P4 _ A)(
simple_P5 _ A)(simple_P6 _ A)(simple_P7 _ A)(simple_P8 _ _ F' F'' F''' A A')2)
E'))with(u(Empty_set _)(sigalg_P1 _ _ F' F'' F''')). 2:apply sym_eq. 2:apply(
u_cmp U F F' F'' F''' u u''''(Intersection _(Empty_set _)E)(Empty_set _)). 2:
exact H. replace(u(Empty_set _)(sigalg_P1 _ F F' F'' F'''))with 0%R.
(**)
replace(0*u(Intersection _(Complement _ A)E)(sigalg_P2 U F F' F'' F'''(
Complement _ A)E(simple_P1 U F F' F'' F''' 2(fun i=>INR i)(fun i=>if eq_nat_dec
i 0 then Complement _ A else if eq_nat_dec i 1 then A else Empty_set _)(
simple_P4 _ A)(simple_P5 _ A)(simple_P6 _ A)(simple_P7 _ A)(simple_P8 _ _ F'
F'' F''' A A')0)E'))%R with 0%R.
(**)
2:apply sym_eq. 2:apply Rmult_eq_0_compat_r. 2:auto. rewrite Rplus_0_l. rewrite
Rmult_1_l. replace((1+1)*0)%R with 0%R. 2:ring. rewrite Rplus_0_r.
(**)
apply(u_cmp U F F' F'' F''' u u''''(Intersection _ A E)A). apply
Extensionality_Ensembles. split. red. intros x H1. elim H1. auto. intuition.
Qed.

(* Specifications: *)
Lemma misc_P0
(U:Type)(f:U->R)(f':forall x:U,(f x>=0)%R)
:forall a:R,(0<a)%R->forall x:U,((fun y:U=>(f y*a))x>=0)%R.