reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S,
  r for Real,
  p for Rational;
reserve X for non empty set,
  f,g for PartFunc of X,REAL,
  r for Real ;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S;
reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  f,g,h for PartFunc of X,REAL,
  A for Element of S,
  r for Real;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL ,
  A,B for Element of S,
  r,s for Real;
reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,REAL,
  r for Real,
  E,A,B for Element of S;

theorem
  f is_integrable_on M implies |. Integral(M,f).| <= Integral(M,abs f)
proof
  assume f is_integrable_on M;
  then R_EAL f is_integrable_on M;
  then |. Integral(M,f) .| <= Integral(M,|. R_EAL f .|) by MESFUNC5:101;
  hence thesis by Th1;
end;
