let a, b, c, d be Real; :: thesis: 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
( f is_integrable_on ['c,d'] & |.f.| is_integrable_on ['c,d'] & |.f.| | ['c,d'] is bounded & |.(integral (f,c,d)).| <= integral (|.f.|,c,d) )
let n be non zero Element of NAT ; :: thesis: 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
( f is_integrable_on ['c,d'] & |.f.| is_integrable_on ['c,d'] & |.f.| | ['c,d'] is bounded & |.(integral (f,c,d)).| <= integral (|.f.|,c,d) )
let f be PartFunc of REAL,(REAL n); :: thesis: ( 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 ( f is_integrable_on ['c,d'] & |.f.| is_integrable_on ['c,d'] & |.f.| | ['c,d'] is bounded & |.(integral (f,c,d)).| <= 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 ) and
A4: ( c in ['a,b'] & d in ['a,b'] ) ; :: thesis: ( f is_integrable_on ['c,d'] & |.f.| is_integrable_on ['c,d'] & |.f.| | ['c,d'] is bounded & |.(integral (f,c,d)).| <= integral (|.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, Th9;
A7: ( ['c,d'] c= dom f & f is_integrable_on ['c,d'] ) by A2, A3, A5, Th9, Th2;
A8: ['a,b'] c= dom |.f.| by A3, NFCONT_4:def 2;
|.f.| | ['a,b'] is bounded by A3, Th19;
then ( ['c,d'] c= dom |.f.| & |.f.| is_integrable_on ['c,d'] ) by A2, A3, A5, A8, INTEGRA6:18;
hence ( f is_integrable_on ['c,d'] & |.f.| is_integrable_on ['c,d'] & |.f.| | ['c,d'] is bounded & |.(integral (f,c,d)).| <= integral (|.f.|,c,d) ) by A2, A6, A7, Th21, Th19; :: thesis: verum