
theorem
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
f be PartFunc of X,ExtREAL, E be Element of S
 st E = dom f & f is E-measurable holds
  Integral(M,f) = Integral(M,max+f) - Integral(M,max-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, E be Element of S;
    assume that
A1:  E = dom f and
A2:  f is E-measurable;
A3: E = dom(max+f) & E = dom(max-f) by A1,MESFUNC2:def 2,def 3;
    max+f is E-measurable & max-f is E-measurable by A1,A2,Th10; then
    Integral(M,max+f) = integral+(M,max+f)
  & Integral(M,max-f) = integral+(M,max-f) by A3,Th5,MESFUNC5:88;
    hence Integral(M,f) = Integral(M,max+f) - Integral(M,max-f)
      by MESFUNC5:def 16;
end;
