(**)
intros a a' x. rewrite<-(Rmult_0_l a). apply Rle_ge. apply
Rmult_le_compat_r. red. auto. apply Rge_le. auto. Qed.

(****************************************************************************** Latter half of this proof adapted from scal_sum in Coq.Reals.PartSum ******************************************************************************)
(*----------------------------------------------------------------------------- For a simple function s and constant c, integral(c*s)=c*integral(s). -----------------------------------------------------------------------------*)
Lemma lebint_P2:
(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)(n
:nat)(v:nat->R)(w:nat->Ensemble U)(S0:simple_0 _ n v w)(S1:simple_1 _ n v w)(S2
:simple_2 _ n v w)(S3:simple_3 _ n v w)(M:msurable _ _ F' F'' F'''(fun x:U=>v(
proj1_sig(simple_P0 _ n v w S0 S1 S2 S3 x))))(E:Ensemble U)(E':F E),
forall(c:R)(c':0<c),lebint_s _ _ _ _ _ u u' u'' u''' u'''' n(fun i:nat=>v i*c)w
(simple_P9 _ n v w S0 S1 S2 S3 c c')(simple_P10 _ n v w S0 S1 S2 S3 c c')(
simple_P11 _ n v w S0 S1 S2 S3 c c')(simple_P12 _ n v w S0 S1 S2 S3 c c')(
simple_P13 _ _ F' F'' F''' n v w S0 S1 S2 S3 M c c') E E' =
c*lebint_s _ _ _ _ _ u u' u'' u''' u'''' n v w S0 S1 S2 S3 M E E')%R.