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 & 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 ; 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); ( 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
; |.(integral (f,d,c)).| <= e * (d - c)
['c,d'] = [.c,d.]
by A2, INTEGRA5:def 3;
then A6:
( |.(integral (f,c,d)).| <= e * (d - c) & integral (f,c,d) = integral (f,['c,d']) )
by A1, A2, A3, A4, A5, Lm14, INTEGR15:19;
['c,d'] = [.c,d.]
by A2, INTEGRA5:def 3;
then
integral (f,d,c) = - (integral (f,['c,d']))
by INTEGR15:20;
hence
|.(integral (f,d,c)).| <= e * (d - c)
by A6, EUCLID:10; verum