let a, b, c, d, e be real number ; :: 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 number st x in ['(min c,d),(max c,d)'] holds
abs (f . x) <= e ) holds
abs (integral f,c,d) <= e * (abs (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 number st x in ['(min c,d),(max c,d)'] holds
abs (f . x) <= e ) implies abs (integral f,c,d) <= e * (abs (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 number st x in ['(min c,d),(max c,d)'] holds
abs (f . x) <= e ; :: thesis: abs (integral f,c,d) <= e * (abs (d - c))
rng (abs f) c= REAL ;
then A3: abs f is Function of (dom (abs f)),REAL by FUNCT_2:4;
['(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_2:38;
A4: vol ['(min c,d),(max c,d)'] = abs (d - c) by Th6;
abs f is_integrable_on ['(min c,d),(max c,d)'] by A1, Th21;
then A5: g is integrable by INTEGRA5:def 2;
reconsider e = e as Real by XREAL_0:def 1;
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

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_1:72;
then A10: g . x = abs (f . x) by VALUED_1:18;
h . x in {e} by A6, A9, FUNCT_2:6;
then h . x = e by TARSKI:def 1;
hence g . x <= h . x by A2, A9, A10; :: thesis: verum

end;

A11: abs (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 5;
(abs f) | ['(min c,d),(max c,d)'] is bounded by A1, Th21;
then A13: g | ['(min c,d),(max c,d)'] is bounded by INTEGRA5:9;
( 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 * (abs (d - c)) by A12, A7, A8, A5, A13, A4, INTEGRA2:34;
hence abs (integral f,c,d) <= e * (abs (d - c)) by A11, XXREAL_0:2; :: thesis: verum