let a, b, c, d, e 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'] & ( for x being Real st x in ['c,d'] holds
|.(f . x).| <= e ) holds
|.(integral (f,d,c)).| <= e * (d - c)
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'] & ( for x being Real st x in ['c,d'] holds
|.(f . x).| <= e ) implies |.(integral (f,d,c)).| <= e * (d - c) )
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'] and
A5: for x being Real st x in ['c,d'] holds
|.(f . x).| <= e ; :: thesis: |.(integral (f,d,c)).| <= e * (d - c)
A6: ( |.(integral (f,c,d)).| <= e * (d - c) & integral (f,c,d) = integral (f,['c,d']) ) by A1, A2, A3, A4, A5, Lm9, INTEGRA5:def 4;

per cases ( c = d or c <> d ) ;

suppose c = d ; :: thesis: |.(integral (f,d,c)).| <= e * (d - c)

hence |.(integral (f,d,c)).| <= e * (d - c) by A1, A3, A5, Lm9; :: thesis: verum

end;

suppose c <> d ; :: thesis: |.(integral (f,d,c)).| <= e * (d - c)

then c < d by A2, XXREAL_0:1;
then integral (f,d,c) = - (integral (f,['c,d'])) by INTEGRA5:def 4;
hence |.(integral (f,d,c)).| <= e * (d - c) by A6, COMPLEX1:52; :: thesis: verum

end;

end;