
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 be Element of S st (ex E be Element of S st E =
dom f & f is E-measurable ) & f is nonnegative & 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 be Element of S;
  assume that
A1: ex E be Element of S st E = dom f & f is E-measurable and
A2: f is nonnegative and
A3: A misses B;
  consider E be Element of S such that
A4: E = dom f and
A5: f is E-measurable by A1;
  ex C be Element of S st C = dom(f|A) & f|A is C-measurable
  proof
    take C=E/\A;
    thus dom(f|A) = C by A4,RELAT_1:61;
A6: C = dom f /\ C by A4,XBOOLE_1:17,28;
A7: dom(f|A) = C by A4,RELAT_1:61
      .= dom(f|C) by A6,RELAT_1:61;
    for x be object st x in dom(f|A) holds (f|A).x = (f|C).x
    proof
      let x be object;
      assume
A8:   x in dom(f|A);
      then (f|A).x = f.x by FUNCT_1:47;
      hence thesis by A7,A8,FUNCT_1:47;
    end;
    then
A9: f|C = f|A by A7,FUNCT_1:2;
    f is C-measurable by A5,MESFUNC1:30,XBOOLE_1:17;
    hence thesis by A6,A9,Th42;
  end;
  then
A10: Integral(M,f|A)=integral+(M,f|A) by A2,Th15,Th88;
  ex C be Element of S st C = dom(f|(A\/B)) & f|(A\/B) is C-measurable
  proof
    reconsider C = E/\(A\/B) as Element of S;
    take C;
    thus dom(f|(A\/B)) = C by A4,RELAT_1:61;
A11: C = dom f /\ C by A4,XBOOLE_1:17,28;
A12: dom(f|(A\/B)) = C by A4,RELAT_1:61
      .= dom(f|C) by A11,RELAT_1:61;
A13: for x be object st x in dom(f|(A\/B)) holds (f|(A\/B)).x = (f|C).x
    proof
      let x be object;
      assume
A14:  x in dom(f|(A\/B));
      then (f|(A\/B)).x = f.x by FUNCT_1:47;
      hence thesis by A12,A14,FUNCT_1:47;
    end;
    f is C-measurable by A5,MESFUNC1:30,XBOOLE_1:17;
    then f|C is C-measurable by A11,Th42;
    hence thesis by A12,A13,FUNCT_1:2;
  end;
  then
A15: Integral(M,f|(A\/B))=integral+(M,f|(A\/B)) by A2,Th15,Th88;
A16: ex C be Element of S st C = dom(f|B) & f|B is C-measurable
  proof
    take C=E/\B;
    thus dom(f|B) = C by A4,RELAT_1:61;
A17: C = dom f /\ C by A4,XBOOLE_1:17,28;
A18: dom(f|B) = C by A4,RELAT_1:61
      .= dom(f|C) by A17,RELAT_1:61;
    for x be object st x in dom(f|B) holds (f|B).x = (f|C).x
    proof
      let x be object;
      assume
A19:  x in dom(f|B);
      then (f|B).x = f.x by FUNCT_1:47;
      hence thesis by A18,A19,FUNCT_1:47;
    end;
    then
A20: f|C = f|B by A18,FUNCT_1:2;
    f is C-measurable by A5,MESFUNC1:30,XBOOLE_1:17;
    hence thesis by A17,A20,Th42;
  end;
  integral+(M,f|(A\/B)) = integral+(M,f|A)+integral+(M,f|B) by A1,A2,A3,Th81;
  hence thesis by A2,A15,A10,A16,Th15,Th88;
end;
