
theorem Th38:
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 f is_integrable_on (COM M) & Integral(M,f) = Integral(COM 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 A1: f is_integrable_on M; then
   consider E be Element of S such that
A2: E = dom f & f is E-measurable by MESFUNC5:def 17;
A3:integral+(M,max+f) < +infty & integral+(M,max-f) < +infty
     by A1,MESFUNC5:def 17;
A4:E = dom(max+f) & E = dom(max-f) by A2,MESFUNC2:def 2,def 3;

   reconsider E1=E as Element of COM(S,M) by Th27;

   max+f is E-measurable & max-f is E-measurable by A2,MESFUN11:10; then
A5:integral+(M,max+f) = integral+(COM M,max+f)
 & integral+(M,max-f) = integral+(COM M,max-f) by A4,Th37,MESFUN11:5;

   f is E1-measurable by A2,Th30;
   hence f is_integrable_on (COM M) by A2,A5,A3,MESFUNC5:def 17;
   Integral(M,f) = integral+(M,max+f) - integral+(M,max-f)
     by MESFUNC5:def 16;
   hence thesis by A5,MESFUNC5:def 16;
end;
