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 Th21:
for f,g being PartFunc of X,ExtREAL
 st f is_integrable_on M & g is_integrable_on M
 holds Integral(M,f+g)
         = Integral(M,f|(dom f /\ dom g)) + Integral(M,g|(dom f /\ dom g))
     & Integral(M,f-g)
         = Integral(M,f|(dom f /\ dom g)) - Integral(M,g|(dom f /\ dom g))
proof
   let f,g be PartFunc of X,ExtREAL;
   assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M;
   consider E be Element of S such that
A3: E = dom f /\ dom g & Integral(M,f+g)=Integral(M,f|E)+Integral(M,g|E)
       by A1,A2,MESFUNC5:109;
   thus Integral(M,f+g)
     = Integral(M,f|(dom f /\ dom g)) + Integral(M,g|(dom f /\ dom g)) by A3;
   ex E0 be Element of S st
    E0 = dom g & g is E0-measurable by A2,MESFUNC5:def 17; then
A4:g is E-measurable by A3,XBOOLE_1:17,MESFUNC1:30;
   ex E be Element of S st
     E = dom f /\ dom g & Integral(M,f-g)=Integral(M,f|E)+Integral(M,(-g)|E)
       by A1,A2,MESFUN10:13; then
   Integral(M,f-g)
    = Integral(M,f|E) + -(Integral(M,g|E)) by A3,A4,XBOOLE_1:17,MESFUN11:55;
   hence Integral(M,f-g)
     = Integral(M,f|(dom f /\ dom g)) - Integral(M,g|(dom f /\ dom g))
       by A3,XXREAL_3:def 4;
end;
