let a, b, c, d, e be Real; :: thesis: for f being PartFunc of REAL,REAL st 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; :: thesis: ( 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 | ['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 ; :: thesis: |.(integral (f,c,d)).| <= e * |.(d - c).|
rng (abs f) c= REAL ;
then A3: abs f is Function of (dom (abs f)),REAL by FUNCT_2:2;
['(min (c,d)),(max (c,d))'] c= dom (abs f) by A1, Th21;
then reconsider g = (abs f) || ['(min (c,d)),(max (c,d))'] as Function of ['(min (c,d)),(max (c,d))'],REAL by A3, FUNCT_2:32;
A4: vol ['(min (c,d)),(max (c,d))'] = |.(d - c).| by Th6;
abs f is_integrable_on ['(min (c,d)),(max (c,d))'] by A1, Th21;
then A5: g is integrable ;
reconsider e = e as Real ;
consider h being Function of ['(min (c,d)),(max (c,d))'],REAL such that
A6: rng h = {e} and
A7: h | ['(min (c,d)),(max (c,d))'] is bounded by INTEGRA4:5;

A8: now :: thesis: for x being Real st x in ['(min (c,d)),(max (c,d))'] holds
g . x <= h . x

let x be Real; :: thesis: ( x in ['(min (c,d)),(max (c,d))'] implies g . x <= h . x )
assume A9: x in ['(min (c,d)),(max (c,d))'] ; :: thesis: g . x <= h . x
then g . x = (abs f) . x by FUNCT_1:49;
then A10: g . x = |.(f . x).| by VALUED_1:18;
h . x in {e} by A6, A9, FUNCT_2:4;
then h . x = e by TARSKI:def 1;
hence g . x <= h . x by A2, A9, A10; :: thesis: verum

end;

A11: |.(integral (f,c,d)).| <= integral ((abs f),(min (c,d)),(max (c,d))) by A1, Th21;
( 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 A12: integral ((abs f),(min (c,d)),(max (c,d))) = integral ((abs f),['(min (c,d)),(max (c,d))']) by INTEGRA5:def 4;
(abs f) | ['(min (c,d)),(max (c,d))'] is bounded by A1, Th21;
then A13: g | ['(min (c,d)),(max (c,d))'] is bounded ;
( h is integrable & integral h = e * (vol ['(min (c,d)),(max (c,d))']) ) by A6, INTEGRA4:4;
then integral ((abs f),(min (c,d)),(max (c,d))) <= e * |.(d - c).| by A12, A7, A8, A5, A13, A4, INTEGRA2:34;
hence |.(integral (f,c,d)).| <= e * |.(d - c).| by A11, XXREAL_0:2; :: thesis: verum