reserve a,b,c,d,e,x,r for Real,
  A for non empty closed_interval Subset of REAL,
  f,g for PartFunc of REAL,REAL;

theorem
  ( 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 be Real
 st x in [' c,d '] holds |.f.x.| <= e ) implies |.integral(f,c,d).| <= e
  * (d-c) & |.integral(f,d,c).| <= e * (d-c) by Lm9,Lm10;
