let a, b, c, d, e 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'] & ( for x being Real st x in ['c,d'] holds
|.(f /. x).| <= e ) holds
|.(integral (f,c,d)).| <= e * (d - c)
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'] & ( 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 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'] & ( 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.| 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, Lm13; :: thesis: verum