
theorem
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,E be Element of S
  st E = dom f & f is E-measurable & f is nonpositive & A misses B
  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,E be Element of S;
    assume that
A1:  E = dom f & f is E-measurable and
A2:  f is nonpositive and
A3:  A misses B;
    set f1 = f|(A\/B);
    set f2 = f|A;
    set f3 = f|B;
    set g = -f;
    reconsider E1 = E /\ (A \/ B) as Element of S;
A4: dom(f|(A\/B)) = E1 by A1,RELAT_1:61;
A5: E1 = dom f /\ E1 by A1,XBOOLE_1:17,28;
A6: f is E1-measurable by A1,XBOOLE_1:17,MESFUNC1:30;
A7: f|E1 = f|E|(A\/B) by RELAT_1:71;
    g|(A\/B) = -(f|(A\/B)) by Th3; then
A8: Integral(M,g|(A\/B)) = -Integral(M,f|(A\/B))
      by A1,A4,A5,A6,A7,Th52,MESFUNC5:42;
    reconsider E2 = E /\ A as Element of S;
A9: dom(f|A) = E2 by A1,RELAT_1:61;
A10:E2 = dom f /\ E2 by A1,XBOOLE_1:17,28;
A11:f is E2-measurable by A1,XBOOLE_1:17,MESFUNC1:30;
A12:f|E2 = f|E|A by RELAT_1:71;
    g|A = -(f|A) by Th3; then
A13:Integral(M,g|A) = - Integral(M,f|A)
      by A1,A9,A10,A11,A12,Th52,MESFUNC5:42;
    reconsider E3 = E /\ B as Element of S;
A14:dom(f|B) = E3 by A1,RELAT_1:61;
A15:E3 = dom f /\ E3 by A1,XBOOLE_1:17,28;
A16:f is E3-measurable by A1,XBOOLE_1:17,MESFUNC1:30;
A17:f|E3 = f|E|B by RELAT_1:71;
    g|B = -(f|B) by Th3; then
A18:Integral(M,g|B) = - Integral(M,f|B)
      by A1,A14,A15,A16,A17,Th52,MESFUNC5:42;
    E = dom g by A1,MESFUNC1:def 7; then
    Integral(M,g|(A\/B)) = Integral(M,g|A) + Integral(M,g|B)
       by A2,A3,A1,MEASUR11:63,MESFUNC5:91; then
    -Integral(M,f|(A\/B)) = -(Integral(M,f|A) + Integral(M,f|B))
       by A8,A13,A18,XXREAL_3:9;
    hence Integral(M,f|(A\/B)) = Integral(M,f|A) + Integral(M,f|B)
       by XXREAL_3:10;
end;
