
theorem
  for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
f be PartFunc of X,ExtREAL st f is_integrable_on M holds |. Integral(M,f) .| <=
  Integral(M,|.f.|)
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL such that
A1: f is_integrable_on M;
A2: |.integral+(M,max+f)-integral+(M,max-f).| <= |.integral+(M,max+f).| +
  |.integral+(M,max-f).| by EXTREAL1:32;
A3: dom f=dom max+f by MESFUNC2:def 2;
A4: now
    let x be object;
    assume x in dom (|.f.|);
    then (|.f.|).x =|. f.x .| by MESFUNC1:def 10;
    hence 0 <= (|.f.|).x by EXTREAL1:14;
  end;
A5: dom f = dom max-f by MESFUNC2:def 3;
A6: |.f.| = max+f + max-f by MESFUNC2:24;
  consider A be Element of S such that
A7: A = dom f and
A8: f is A-measurable by A1;
A9: max-f is A-measurable by A7,A8,MESFUNC2:26;
A10: max+f is nonnegative by Lm1;
  then 0 <= integral+(M,max+f) by A7,A8,A3,Th79,MESFUNC2:25;
  then
A11: |.Integral(M,f).| <= integral+(M,max+f) + |.integral+(M,max-f).| by A2,
EXTREAL1:def 1;
A12: max+f is A-measurable by A8,MESFUNC2:25;
A13: A = dom |.f.| by A7,MESFUNC1:def 10;
A14: max-f is nonnegative by Lm1;
  then
A15: 0 <= integral+(M,max-f) by A7,A8,A5,Th79,MESFUNC2:26;
  |.f.| is A-measurable by A7,A8,MESFUNC2:27;
  then Integral(M,|.f.|) = integral+(M,max+f + max-f) by A13,A4,A6,Th88,
SUPINF_2:52
    .= integral+(M,max+f)+integral+(M,max-f) by A7,A3,A5,A10,A14,A12,A9,Lm10;
  hence thesis by A15,A11,EXTREAL1:def 1;
end;
