let a, b, c, d, e be Real; :: thesis: for n being non zero Element of NAT
for f being PartFunc of REAL,(REAL n) st a <= b & c <= d & 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 ['c,d'] holds
|.(f /. x).| <= e ) holds
|.(integral (f,d,c)).| <= e * (d - c)

let n be non zero Element of NAT ; :: thesis: for f being PartFunc of REAL,(REAL n) st a <= b & c <= d & 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 ['c,d'] holds
|.(f /. x).| <= e ) holds
|.(integral (f,d,c)).| <= e * (d - c)

let f be PartFunc of REAL,(REAL n); :: thesis: ( a <= b & c <= d & 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 ['c,d'] holds
|.(f /. x).| <= e ) implies |.(integral (f,d,c)).| <= e * (d - c) )

assume that
A1: a <= b and
A2: c <= d and
A3: ( 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'] ) and
A4: d in ['a,b'] and
A5: for x being Real st x in ['c,d'] holds
|.(f /. x).| <= e ; :: thesis: |.(integral (f,d,c)).| <= e * (d - c)
['c,d'] = [.c,d.] by ;
then A6: ( |.(integral (f,c,d)).| <= e * (d - c) & integral (f,c,d) = integral (f,['c,d']) ) by ;
['c,d'] = [.c,d.] by ;
then integral (f,d,c) = - (integral (f,['c,d'])) by INTEGR15:20;
hence |.(integral (f,d,c)).| <= e * (d - c) by ; :: thesis: verum