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 A1, A2, A3, A4, Lm6, INTEGRA5:def 4;

|.(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 A1, A2, A3, A4, Lm6, INTEGRA5:def 4;