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
|.(integral (f,d,c)).| <= 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 |.(integral (f,d,c)).| <= 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 & c in ['a,b'] ) and
A4: d in ['a,b'] ; :: thesis: |.(integral (f,d,c)).| <= integral ((abs f),c,d)
A5: ( |.(integral (f,c,d)).| <= integral ((abs f),c,d) & integral (f,c,d) = integral (f,['c,d']) ) by ;
per cases ( c = d or c <> d ) ;
suppose c = d ; :: thesis: |.(integral (f,d,c)).| <= integral ((abs f),c,d)
hence |.(integral (f,d,c)).| <= integral ((abs f),c,d) by A1, A3, Lm6; :: thesis: verum
end;
suppose c <> d ; :: thesis: |.(integral (f,d,c)).| <= integral ((abs f),c,d)
then c < d by ;
then integral (f,d,c) = - (integral (f,['c,d'])) by INTEGRA5:def 4;
hence |.(integral (f,d,c)).| <= integral ((abs f),c,d) by ; :: thesis: verum
end;
end;