
theorem
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
f be nonnegative PartFunc of X,ExtREAL, E be Element of S
  st E = dom f & f is E-measurable
  holds Integral(M,max-f) = 0 & integral+(M,max-f) = 0
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    f be nonnegative PartFunc of X,ExtREAL, E be Element of S;
    assume that
A1:  E = dom f and
A2:  f is E-measurable;
A3: E = dom(max-f) by A1,MESFUNC2:def 3;
A4: max-f is E-measurable by A1,A2,Th10;
    for x be object st x in dom(max-f) holds (max-f).x = 0
    proof
     let x be object;
     assume A5: x in dom(max-f); then
A6:  x in dom(max+f) by A1,A3,MESFUNC2:def 2;
     per cases by SUPINF_2:51;
     suppose f.x = 0; then
      (max+f).x = f.x by A5,MESFUNC2:18;
      hence (max-f).x = 0 by A5,MESFUNC2:19;
     end;
     suppose f.x > 0; then
      max(f.x,0) = f.x by XXREAL_0:def 10; then
      (max+f).x = f.x by A6,MESFUNC2:def 2;
      hence (max-f).x = 0 by A5,MESFUNC2:19;
     end;
    end;
    hence Integral(M,max-f) = 0 * M.(dom(max-f)) by A3,MEASUR10:27 .= 0;
    hence integral+(M,max-f) = 0 by A3,A4,Th5,MESFUNC5:88;
end;
