let a, b, c, d, e be real number ; :: 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 number st x in ['c,d'] holds
abs (f . x) <= e ) holds
abs (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 number st x in ['c,d'] holds
abs (f . x) <= e ) implies abs (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 number st x in ['c,d'] holds
abs (f . x) <= e ) ) ; :: thesis: abs (integral f,c,d) <= e * (d - c)
0 <= d - c by A2, XREAL_1:50;
then A4: abs (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 abs (integral f,c,d) <= e * (d - c) by A1, A3, A4, Lm8; :: thesis: verum