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 Th7:
  A c= dom f & f is_integrable_on A & f|A is bounded implies abs f
  is_integrable_on A & |.integral(f,A).| <= integral(abs f,A)
proof
A1: |.f||A.| = abs(f)||A by RFUNCT_1:46;
  assume A c= dom f;
  then
A2: f||A is Function of A,REAL by Lm1;
  assume f is_integrable_on A & f|A is bounded;
  then
A3: f||A is integrable & f||A|A is bounded;
  thus thesis by A3,A2,A1,INTEGRA4:23;
end;
