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,c,d)).| <= 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,c,d)).| <= 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'] & d in ['a,b'] & ( for x being Real st x in ['c,d'] holds

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

0 <= d - c by A2, XREAL_1:48;

then A4: |.(d - c).| = d - c by ABSVALUE:def 1;

( min (c,d) = c & max (c,d) = d ) by A2, XXREAL_0:def 9, XXREAL_0:def 10;

hence |.(integral (f,c,d)).| <= e * (d - c) by A1, A3, A4, Lm8; :: thesis: verum

|.(f . x).| <= e ) holds

|.(integral (f,c,d)).| <= 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,c,d)).| <= 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'] & d in ['a,b'] & ( for x being Real st x in ['c,d'] holds

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

0 <= d - c by A2, XREAL_1:48;

then A4: |.(d - c).| = d - c by ABSVALUE:def 1;

( min (c,d) = c & max (c,d) = d ) by A2, XXREAL_0:def 9, XXREAL_0:def 10;

hence |.(integral (f,c,d)).| <= e * (d - c) by A1, A3, A4, Lm8; :: thesis: verum