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 Th8:
  a<=b & [' a,b '] c= dom f & f is_integrable_on [' a,b '] & f|[' a
  ,b '] is bounded implies |.integral(f,a,b).| <=integral(abs f,a,b)
proof
  assume a<=b;
  then
  integral(f,a,b) = integral(f,[' a,b ']) & integral(abs f,a,b) = integral
  (abs f,[' a,b ']) by INTEGRA5:def 4;
  hence thesis by Th7;
end;
