reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  c for Complex,
  E,A,B for Element of S;

theorem
  ( ex A be Element of S st A = dom f & f is A-measurable ) & f
  is_integrable_on M implies |. Integral(M,f).| <= Integral(M,|.f.|)
proof
  assume that
A1: ex A be Element of S st A = dom f & f is A-measurable and
A2: f is_integrable_on M;
  per cases;
  suppose
    Integral(M,f) = 0;
    hence thesis by A1,A2,Lm2;
  end;
  suppose
    Integral(M,f) <> 0;
    hence thesis by A1,A2,Lm3;
  end;
end;
