let a, b, c, d be Real; for f being PartFunc of REAL,REAL st a <= b & c <= d & 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
( ['c,d'] c= dom (abs f) & abs f is_integrable_on ['c,d'] & (abs f) | ['c,d'] is bounded & |.(integral (f,c,d)).| <= integral ((abs f),c,d) )
let f be PartFunc of REAL,REAL; ( a <= b & c <= d & 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 ( ['c,d'] c= dom (abs f) & abs f is_integrable_on ['c,d'] & (abs f) | ['c,d'] is bounded & |.(integral (f,c,d)).| <= integral ((abs f),c,d) ) )
assume that
A1:
a <= b
and
A2:
c <= d
and
A3:
( f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f )
and
A4:
( c in ['a,b'] & d in ['a,b'] )
; ( ['c,d'] c= dom (abs f) & abs f is_integrable_on ['c,d'] & (abs f) | ['c,d'] is bounded & |.(integral (f,c,d)).| <= integral ((abs f),c,d) )
['a,b'] = [.a,b.]
by A1, INTEGRA5:def 3;
then A5:
( a <= c & d <= b )
by A4, XXREAL_1:1;
then A6:
f | ['c,d'] is bounded
by A2, A3, Th18;
( ['c,d'] c= dom f & f is_integrable_on ['c,d'] )
by A2, A3, A5, Th18;
hence
( ['c,d'] c= dom (abs f) & abs f is_integrable_on ['c,d'] & (abs f) | ['c,d'] is bounded & |.(integral (f,c,d)).| <= integral ((abs f),c,d) )
by A2, A6, Th7, Th8, RFUNCT_1:82, VALUED_1:def 11; verum