let a, b, c, d, e be Real; for n being non zero Element of NAT
for f being PartFunc of REAL,(REAL n) st a <= b & f is_integrable_on ['a,b'] & |.f.| is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] & ( for x being Real st x in ['(min (c,d)),(max (c,d))'] holds
|.(f /. x).| <= e ) holds
|.(integral (f,c,d)).| <= e * |.(d - c).|
let n be non zero Element of NAT ; for f being PartFunc of REAL,(REAL n) st a <= b & f is_integrable_on ['a,b'] & |.f.| is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] & ( for x being Real st x in ['(min (c,d)),(max (c,d))'] holds
|.(f /. x).| <= e ) holds
|.(integral (f,c,d)).| <= e * |.(d - c).|
let f be PartFunc of REAL,(REAL n); ( a <= b & f is_integrable_on ['a,b'] & |.f.| is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] & ( for x being Real st x in ['(min (c,d)),(max (c,d))'] holds
|.(f /. x).| <= e ) implies |.(integral (f,c,d)).| <= e * |.(d - c).| )
set A = ['(min (c,d)),(max (c,d))'];
assume that
A1:
( a <= b & f is_integrable_on ['a,b'] & |.f.| is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] )
and
A2:
for x being Real st x in ['(min (c,d)),(max (c,d))'] holds
|.(f /. x).| <= e
; |.(integral (f,c,d)).| <= e * |.(d - c).|
rng |.f.| c= REAL
;
then A3:
|.f.| is Function of (dom |.f.|),REAL
by FUNCT_2:2;
dom |.f.| = dom f
by NFCONT_4:def 2;
then A4:
['(min (c,d)),(max (c,d))'] c= dom |.f.|
by A1, Th3;
then reconsider g = |.f.| | ['(min (c,d)),(max (c,d))'] as Function of ['(min (c,d)),(max (c,d))'],REAL by A3, FUNCT_2:32;
A5:
vol ['(min (c,d)),(max (c,d))'] = |.(d - c).|
by INTEGRA6:6;
|.f.| is_integrable_on ['(min (c,d)),(max (c,d))']
by A1, Th22;
then A6:
g is integrable
;
consider h being Function of ['(min (c,d)),(max (c,d))'],REAL such that
A7:
rng h = {e}
and
A8:
h | ['(min (c,d)),(max (c,d))'] is bounded
by INTEGRA4:5;
A9:
now for x being Real st x in ['(min (c,d)),(max (c,d))'] holds
g . x <= h . xlet x be
Real;
( x in ['(min (c,d)),(max (c,d))'] implies g . x <= h . x )assume A10:
x in ['(min (c,d)),(max (c,d))']
;
g . x <= h . xthen
g . x = |.f.| . x
by FUNCT_1:49;
then
g . x = |.f.| /. x
by A10, A4, PARTFUN1:def 6;
then A11:
g . x = |.(f /. x).|
by A10, A4, NFCONT_4:def 2;
h . x in {e}
by A7, A10, FUNCT_2:4;
then
h . x = e
by TARSKI:def 1;
hence
g . x <= h . x
by A11, A2, A10;
verum end;
A12:
|.(integral (f,c,d)).| <= integral (|.f.|,(min (c,d)),(max (c,d)))
by A1, Th22;
( min (c,d) <= c & c <= max (c,d) )
by XXREAL_0:17, XXREAL_0:25;
then
min (c,d) <= max (c,d)
by XXREAL_0:2;
then A13:
integral (|.f.|,(min (c,d)),(max (c,d))) = integral (|.f.|,['(min (c,d)),(max (c,d))'])
by INTEGRA5:def 4;
|.f.| | ['(min (c,d)),(max (c,d))'] is bounded
by A1, Th22;
then A14:
g | ['(min (c,d)),(max (c,d))'] is bounded
;
( h is integrable & integral h = e * (vol ['(min (c,d)),(max (c,d))']) )
by A7, INTEGRA4:4;
then
integral (|.f.|,(min (c,d)),(max (c,d))) <= e * |.(d - c).|
by A13, A8, A9, A6, A14, A5, INTEGRA2:34;
hence
|.(integral (f,c,d)).| <= e * |.(d - c).|
by A12, XXREAL_0:2; verum