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 '] implies [' c,d
  '] c= dom (abs f) & abs f is_integrable_on [' c,d '] & abs f|[' c,d '] is
  bounded & |.integral(f,c,d).| <= integral(abs f,c,d) & |.integral(f,d,c).| <=
  integral(abs f,c,d) by Lm6,Lm7;
