let a, b, c, d be Real; :: 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 & |.(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 & |.(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'] ) ; :: thesis: ( ['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; :: thesis: verum

( ['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; :: 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 & |.(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'] ) ; :: thesis: ( ['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; :: thesis: verum