
theorem Th18:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 f be PartFunc of X,ExtREAL, A,B be Element of S
 st A \/ B c= dom f & f is (A\/B)-measurable & A misses B
  & (integral+(M,max+(f|(A\/B))) < +infty
   or integral+(M,max-(f|(A\/B))) < +infty)
 holds Integral(M,f|(A\/B)) = Integral(M,f|A)+Integral(M,f|B)
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
     f be PartFunc of X,ExtREAL, A,B be Element of S;
    assume that
A1:  A \/ B c= dom f and
A2:  f is (A\/B)-measurable and
A3:  A misses B and
A4:  integral+(M,max+(f|(A\/B))) < +infty
     or integral+(M,max-(f|(A\/B))) < +infty;

    set g=f|(A\/B);
A5: dom g = A \/ B by A1,RELAT_1:62;

    A \/ B = dom f /\ (A \/ B) by A1,XBOOLE_1:28; then
A6: g is (A\/B)-measurable by A2,MESFUNC5:42;

    g|A = f|A & g|B = f|B by RELAT_1:74,XBOOLE_1:7;
    hence Integral(M,f|(A\/B)) = Integral(M,f|A)+Integral(M,f|B)
      by A6,A5,A3,A4,Lm2;
end;
