
theorem Th88:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure
  of S, f be PartFunc of X,ExtREAL st (ex A be Element of S st A = dom f & f
  is A-measurable) & f is nonnegative 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;
  assume that
A1: ex A be Element of S st A = dom f & f is A-measurable and
A2: f is nonnegative;
A3: dom f = dom max+ f by MESFUNC2:def 2;
A4: now
    let x be object;
A5: 0 <= f.x by A2,SUPINF_2:51;
    assume x in dom f;
    hence max+f.x = max(f.x,0) by A3,MESFUNC2:def 2
      .=f.x by A5,XXREAL_0:def 10;
  end;
A6: dom f = dom max-f by MESFUNC2:def 3;
A7: now
    let x be Element of X;
    assume x in dom max- f;
    then max+f.x=f.x by A4,A6;
    hence max-f.x=0 by MESFUNC2:19;
  end;
A8: dom f=dom (max- f) by MESFUNC2:def 3;
  f = max+ f by A3,A4,FUNCT_1:2;
  hence Integral(M,f) =integral+(M,f) - 0 by A1,A7,A8,Th87,MESFUNC2:26
    .=integral+(M,f) by XXREAL_3:15;
end;
