let a, b, c, d 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'] holds
|.(integral (f,d,c)).| <= integral (|.f.|,c,d)
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'] holds
|.(integral (f,d,c)).| <= integral (|.f.|,c,d)
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'] implies |.(integral (f,d,c)).| <= integral (|.f.|,c,d) )
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']
; |.(integral (f,d,c)).| <= integral (|.f.|,c,d)
['c,d'] = [.c,d.]
by A2, INTEGRA5:def 3;
then A5:
( |.(integral (f,c,d)).| <= integral (|.f.|,c,d) & integral (f,c,d) = integral (f,['c,d']) )
by A1, A2, A3, A4, Lm11, 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)).| <= integral (|.f.|,c,d)
by A5, EUCLID:10; verum