let a, b, c, d be real number ; :: thesis: 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 & abs (integral f,c,d) <= integral (abs f),c,d )
let f be PartFunc of REAL , REAL ; :: thesis: ( 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 & abs (integral f,c,d) <= integral (abs f),c,d ) )
assume A1: ( 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'] ) ; :: thesis: ( ['c,d'] c= dom (abs f) & abs f is_integrable_on ['c,d'] & (abs f) | ['c,d'] is bounded & abs (integral f,c,d) <= integral (abs f),c,d )
['a,b'] = [.a,b.] by A1, INTEGRA5:def 4;
then ( a <= c & d <= b ) by A1, XXREAL_1:1;
then ( ['c,d'] c= dom f & f is_integrable_on ['c,d'] & f | ['c,d'] is bounded ) by A1, Th18;
hence ( ['c,d'] c= dom (abs f) & abs f is_integrable_on ['c,d'] & (abs f) | ['c,d'] is bounded & abs (integral f,c,d) <= integral (abs f),c,d ) by A1, Th7, Th8, RFUNCT_1:99, VALUED_1:def 11; :: thesis: verum