
theorem Th46:
  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
 ) & f"{+infty} in S & f"{-infty} in S & g"{+infty} in S & g"{-infty} in S
  holds dom(f+g) in S
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 and
A3: f"{+infty} in S and
A4: f"{-infty} in S and
A5: g"{+infty} in S and
A6: g"{-infty} in S;
A7: f"{+infty} /\ g"{-infty} in S by A3,A6,MEASURE1:34;
  f"{-infty} /\ g"{+infty} in S by A4,A5,MEASURE1:34;
  then f"{-infty}/\g"{+infty} \/ f"{+infty}/\g"{-infty} in S by A7,MEASURE1:34;
  then
A8: X \ (f"{-infty}/\g"{+infty} \/ f"{+infty}/\g"{-infty}) in S by MEASURE1:34;
  consider E2 be Element of S such that
A9: E2=dom g and
  g is E2-measurable by A2;
  consider E1 be Element of S such that
A10: E1=dom f and
  f is E1-measurable by A1;
A11: E1/\E2/\(X \ (f"{-infty}/\g"{+infty} \/ f"{+infty}/\g"{-infty})) = (E1
  /\E2/\X) \ (f"{-infty}/\g"{+infty} \/ f"{+infty}/\g"{-infty}) by XBOOLE_1:49
    .=(E1/\E2) \ (f"{-infty}/\g"{+infty} \/ f"{+infty}/\g"{-infty}) by
XBOOLE_1:28;
  dom(f+g) = (E1 /\ E2) \(f"{-infty}/\g"{+infty} \/ f"{+infty}/\g"{-infty}
  ) by A10,A9,MESFUNC1:def 3;
  hence thesis by A8,A11,MEASURE1:34;
end;
