
theorem Th47:
  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 (ex E1 be Element of S st E1=dom f & f
is E1-measurable) & (ex E2 be Element of S st E2=dom g & g is E2-measurable
  ) holds ex E be Element of S st E=dom(f+g) & (f+g) is E-measurable
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: ex E1 be Element of S st E1=dom f & f is E1-measurable and
A2: ex E2 be Element of S st E2=dom g & g is E2-measurable;
  consider E1 being Element of S such that
A3: E1 = dom f and
A4: f is E1-measurable by A1;
  consider E2 being Element of S such that
A5: E2 = dom g and
A6: g is E2-measurable by A2;
  set E3 = E1 /\ E2;
  set g1 = g|E3;
A7: g1"{-infty} = E3 /\ g"{-infty} by FUNCT_1:70;
  set f1 = f|E3;
  dom f1 = dom f /\ E3 by RELAT_1:61;
  then
A8: dom f1 = E3 by A3,XBOOLE_1:17,28;
  g is E3-measurable by A6,MESFUNC1:30,XBOOLE_1:17;
  then
A9: g1 is E3-measurable by Lm6;
A10: g1"{+infty} = E3 /\ g"{+infty} by FUNCT_1:70;
  dom g1 = dom g /\ E3 by RELAT_1:61;
  then
A11: dom g1 = E3 by A5,XBOOLE_1:17,28;
  f1"{+infty} = E3 /\ f"{+infty} by FUNCT_1:70;
  then
A12: f1"{+infty} /\ g1"{-infty} = f"{+infty} /\ (E3 /\ (E3 /\ g"{-infty}))
  by A7,XBOOLE_1:16
    .= f"{+infty} /\ (E3 /\ E3 /\ g"{-infty}) by XBOOLE_1:16
    .= f"{+infty} /\ g"{-infty} /\ E3 by XBOOLE_1:16;
A13: dom(f1+g1) = (dom f1 /\ dom g1) \ (f1"{-infty}/\g1"{+infty} \/ f1"{
  +infty}/\g1"{-infty}) by MESFUNC1:def 3;
  f is E3-measurable by A4,MESFUNC1:30,XBOOLE_1:17;
  then f1 is E3-measurable by Lm6;
  then consider E be Element of S such that
A14: E = dom(f1+g1) and
A15: (f1+g1) is E-measurable by A9,A8,A11,Lm7;
  take E;
A16: dom((f+g)|E) = dom(f+g) /\ E by RELAT_1:61;
  f1"{-infty} = E3 /\ f"{-infty} by FUNCT_1:70;
  then f1"{-infty} /\ g1"{+infty} = f"{-infty} /\ (E3 /\ (E3 /\ g"{+infty}))
  by A10,XBOOLE_1:16
    .= f"{-infty} /\ (E3 /\ E3 /\ g"{+infty}) by XBOOLE_1:16
    .= f"{-infty} /\ g"{+infty} /\ E3 by XBOOLE_1:16;
  then
A17: f1"{-infty}/\g1"{+infty} \/ f1"{+infty}/\g1"{-infty} = E3 /\ (f"{-infty
  }/\g"{+infty} \/ f"{+infty}/\g"{-infty}) by A12,XBOOLE_1:23;
A18: dom(f+g) = (dom f /\ dom g) \ (f"{-infty}/\g"{+infty} \/ f"{+infty}/\g"
  {-infty}) by MESFUNC1:def 3;
  then
A19: dom(f+g) = E by A3,A5,A8,A11,A14,A13,A17,XBOOLE_1:47;
  now
    let v be Element of X;
    assume
A20: v in dom((f+g)|E);
    then
A21: v in dom(f+g) /\ E by RELAT_1:61;
    then
A22: v in dom(f+g) by XBOOLE_0:def 4;
A23: ((f+g)|E).v = (f+g).v by A20,FUNCT_1:47
      .= f.v + g.v by A22,MESFUNC1:def 3;
A24: v in E by A21,XBOOLE_0:def 4;
A25: E c= E3 by A8,A11,A14,A13,XBOOLE_1:36;
    (f1+g1).v = f1.v + g1.v by A14,A19,A16,A20,MESFUNC1:def 3
      .= f.v + g1.v by A8,A24,A25,FUNCT_1:47;
    hence ((f+g)|E).v = (f1+g1).v by A11,A24,A25,A23,FUNCT_1:47;
  end;
  then (f+g)|E = f1+g1 by A14,A19,A16,PARTFUN1:5;
  hence thesis by A3,A5,A8,A11,A14,A15,A13,A17,A18,Lm6,XBOOLE_1:47;
end;
