reserve X for set;
reserve n,i for Element of NAT;
reserve a,b,c,d,e,r,x0 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve f,g,h for PartFunc of REAL,REAL n;
reserve E for Element of REAL n;

theorem
  for n be non zero Element of NAT
  for f be 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']
  holds |.f.| is_integrable_on ['c,d'] &
  |.f.| | ['c,d'] is bounded & |.integral(f,c,d).| <= integral(|.f.|,c,d) &
  |.integral(f,d,c).| <= integral(|.f.|,c,d) by Lm11,Lm12;
