(**)
intros. unfold lebint_s. cut(forall(An Bn:nat->R)(N:nat)(x:R),(forall i,Bn i=(x
*An i)%R)->(x*sum_f_R0 An N)%R=sum_f_R0 Bn N). intro H. apply sym_eq. apply H.
(**)
intro. rewrite Rmult_assoc. rewrite Rmult_comm. rewrite Rmult_assoc. apply
Rmult_eq_compat_l. rewrite Rmult_comm. apply Rmult_eq_compat_l. apply(u_cmp U F
F' F'' F''' u u''''). auto.
(**)
intros An Bn N x H. induction N as[|N HrecN]. simpl. auto. do 2 rewrite tech5.
rewrite Rmult_plus_distr_l. rewrite<-HrecN. apply Rplus_eq_compat_l. apply
sym_eq. apply H. Qed.

(* Specifications: *)
Theorem markov:
(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)(f'':msurable _ _ F' F'' F''' f)(a:R)(a':0<a),
forall L,lebint _ _ _ _ _ u u' u'' u''' u'''' f f' f''(Full_set U)(sigalg_P3 U
F F' F'' F''') L
->u(fun x:U=>f x>=a)(msurable_P1 _ _ F' F'' F''' f f' f'' a a')<=/a*L)%R.