
theorem
  for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
  f be PartFunc of X,ExtREAL, A be Element of S st (ex E be Element of S st E =
  dom f & f is E-measurable) & M.A = 0 holds Integral(M,f|A)=0
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL, A be Element of S such that
A1: ex E be Element of S st E = dom f & f is E-measurable and
A2: M.A = 0;
A3: dom f=dom (max+ f) by MESFUNC2:def 2;
  max+f is nonnegative by Lm1;
  then
A4: integral+(M,(max+ f)|A)=0 by A1,A2,A3,Th82,MESFUNC2:25;
A5: dom f=dom (max- f) by MESFUNC2:def 3;
A6: max-f is nonnegative by Lm1;
  Integral(M,f|A) = integral+(M,(max+ f)|A) - integral+(M,max-(f|A)) by Th28
    .= integral+(M,(max+ f)|A) - integral+(M,(max- f)|A) by Th28
    .= 0.- 0. by A1,A2,A5,A6,A4,Th82,MESFUNC2:26;
  hence thesis;

end;
