reserve X for set;
reserve X,X1,X2 for non empty set;
reserve S for SigmaField of X;
reserve S1 for SigmaField of X1;
reserve S2 for SigmaField of X2;
reserve M for sigma_Measure of S;
reserve M1 for sigma_Measure of S1;
reserve M2 for sigma_Measure of S2;

theorem
for f being PartFunc of X,ExtREAL holds f is_integrable_on M iff
    max+f is_integrable_on M & max-f is_integrable_on M
proof
    let f be PartFunc of X,ExtREAL;
    hereby 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;
A5:  max+f is E-measurable & max-f is E-measurable by A2,MESFUN11:10;
A6:  max+f is nonnegative & max-f is nonnegative by MESFUN11:5; then
A7:  integral+(M,max+(max+f)) < +infty
   & integral+(M,max+(max-f)) < +infty by A3,MESFUN11:31;
     integral+(M,max-(max+f)) < +infty
   & integral+(M,max-(max-f)) < +infty by A4,A5,A6,MESFUN11:53;
     hence max+f is_integrable_on M & max-f is_integrable_on M
      by A4,A5,A7,MESFUNC5:def 17;
    end;
    assume that
A8:  max+f is_integrable_on M and
A9:  max-f is_integrable_on M;
    consider E1 be Element of S such that
A10: E1 = dom(max+f) & max+f is E1-measurable by A8,MESFUNC5:def 17;
    consider E2 be Element of S such that
A11: E2 = dom(max-f) & max-f is E2-measurable by A9,MESFUNC5:def 17;
A12:E1 = dom f by A10,MESFUNC2:def 2; then
    E1 = E2 by A11,MESFUNC2:def 3; then
A13:f is E1-measurable by A10,A11,A12,MESFUN11:10;
    max+f is nonnegative & max-f is nonnegative by MESFUN11:5; then
    max+(max+f) = max+f & max+(max-f) = max-f by MESFUN11:31; then
    integral+(M,max+f) < +infty & integral+(M,max-f) < +infty
      by A8,A9,MESFUNC5:def 17;
    hence f is_integrable_on M by A12,A13,MESFUNC5:def 17;
end;
