
theorem
  for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
  f,g be PartFunc of X,ExtREAL st f is_integrable_on M & g is_integrable_on M
holds ex E be Element of S st E = dom f /\ dom g & Integral(M,f+g)=Integral(M,f
  |E)+Integral(M,g|E)
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g
  be PartFunc of X,ExtREAL;
  assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M;
  consider B be Element of S such that
A3: B = dom g and
  g is B-measurable by A2;
  consider A be Element of S such that
A4: A = dom f and
  f is A-measurable by A1;
  set E = A /\ B;
  set g1 = g|E;
  set f1 = f|E;
  take E = A /\ B;
A5: dom f1 = dom f /\ (A/\B) by RELAT_1:61
    .= A /\ A /\ B by A4,XBOOLE_1:16;
A6: f1"{+infty} = E /\ (f"{+infty}) by FUNCT_1:70;
  g1"{-infty} = E /\ (g"{-infty}) by FUNCT_1:70;
  then
A7: f1"{+infty} /\ g1"{-infty} = f"{+infty} /\ E /\ E /\ g"{-infty} by A6,
XBOOLE_1:16
    .= f"{+infty} /\ (E /\ E) /\ g"{-infty} by XBOOLE_1:16
    .= E /\ (f"{+infty} /\ g"{-infty}) by XBOOLE_1:16;
A8: g1"{+infty} = E /\ (g"{+infty}) by FUNCT_1:70;
  f1"{-infty} = E /\ (f"{-infty}) by FUNCT_1:70;
  then f1"{-infty} /\ g1"{+infty} = f"{-infty} /\ E /\ E /\ g"{+infty} by A8,
XBOOLE_1:16
    .= f"{-infty} /\ (E /\ E) /\ g"{+infty} by XBOOLE_1:16
    .= E /\ (f"{-infty} /\ g"{+infty}) by XBOOLE_1:16;
  then
A9: f1"{-infty}/\g1"{+infty} \/ f1"{+infty}/\g1"{-infty} = E /\ (f"{-infty}
  /\g"{+infty} \/ f"{+infty}/\g"{-infty}) by A7,XBOOLE_1:23;
A10: dom g1 = dom g /\ (A/\B) by RELAT_1:61
    .= B /\ B /\ A by A3,XBOOLE_1:16;
A11: dom(f1+g1) = (dom f1 /\ dom g1) \ (f1"{-infty}/\g1"{+infty} \/ f1"{
  +infty}/\g1"{-infty}) by MESFUNC1:def 3
    .= E \ (f"{-infty}/\g"{+infty} \/ f"{+infty}/\g"{-infty}) by A5,A10,A9,
XBOOLE_1:47
    .= dom(f+g) by A4,A3,MESFUNC1:def 3;
A12: for x be object st x in dom(f1+g1) holds (f1+g1).x = (f+g).x
  proof
    let x be object;
    assume
A13: x in dom(f1+g1);
    then x in (dom f1 /\ dom g1) \ (f1"{-infty} /\ g1"{+infty} \/ f1"{+infty}
    /\g1"{-infty}) by MESFUNC1:def 3;
    then
A14: x in dom f1 /\ dom g1 by XBOOLE_0:def 5;
    then
A15: x in dom f1 by XBOOLE_0:def 4;
A16: x in dom g1 by A14,XBOOLE_0:def 4;
    (f1+g1).x = f1.x + g1.x by A13,MESFUNC1:def 3
      .= f.x + g1.x by A15,FUNCT_1:47
      .= f.x + g.x by A16,FUNCT_1:47;
    hence thesis by A11,A13,MESFUNC1:def 3;
  end;
  thus E = dom f /\ dom g by A4,A3;
A17: g1 is_integrable_on M by A2,Th97;
  f1 is_integrable_on M by A1,Th97;
  then Integral(M,f1+g1) = Integral(M,f1) + Integral(M,g1) by A17,A5,A10,Lm13;
  hence thesis by A11,A12,FUNCT_1:2;
end;
