
theorem Th78:
  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 A be Element of S st A = dom f & f
is A-measurable ) & ( ex B be Element of S st B = dom g & g is B-measurable
) & f is nonnegative & g is nonnegative holds ex C be Element of S st C = dom
  (f+g) & integral+(M,f+g) = integral+(M,f|C) + integral+(M,g|C)
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 A be Element of S st A = dom f & f is A-measurable and
A2: ex B be Element of S st B = dom g & g is B-measurable and
A3: f is nonnegative and
A4: g is nonnegative;
  set g1 = g|(dom f /\ dom g);
A5: g1 is without-infty by A4,Th12,Th15;
A6: g1 is nonnegative by A4,Th15;
  dom g1 = dom g /\ (dom f /\ dom g) by RELAT_1:61;
  then
A7: dom g1 = dom g /\ dom g /\ dom f by XBOOLE_1:16;
  consider B be Element of S such that
A8: B = dom g and
A9: g is B-measurable by A2;
  consider A be Element of S such that
A10: A = dom f and
A11: f is A-measurable by A1;
  take C = A /\ B;
A12: C = dom(f+g) by A3,A4,A10,A8,Th16;
A13: C = dom g /\ C by A8,XBOOLE_1:17,28;
  g is C-measurable by A9,MESFUNC1:30,XBOOLE_1:17;
  then
A14: g|C is C-measurable by A13,Th42;
A15: C = dom f /\ C by A10,XBOOLE_1:17,28;
  f is C-measurable by A11,MESFUNC1:30,XBOOLE_1:17;
  then
A16: f|C is C-measurable by A15,Th42;
  set f1 = f|(dom f /\ dom g);
  dom f1 = dom f /\ (dom f /\ dom g) by RELAT_1:61;
  then
A17: dom f1 = dom f /\ dom f /\ dom g by XBOOLE_1:16;
A18: f1 is without-infty by A3,Th12,Th15;
  then
A19: dom(f1+g1) = C /\ C by A10,A8,A17,A7,A5,Th16;
A20: dom(f1+g1) = dom f1 /\ dom g1 by A18,A5,Th16;
A21: for x be object st x in dom(f1+g1) holds (f1+g1).x = (f+g).x
  proof
    let x be object;
    assume
A22: x in dom(f1+g1);
    then
A23: x in dom f1 by A20,XBOOLE_0:def 4;
A24: x in dom g1 by A20,A22,XBOOLE_0:def 4;
    (f1+g1).x = f1.x + g1.x by A22,MESFUNC1:def 3
      .= f.x + g1.x by A23,FUNCT_1:47
      .= f.x + g.x by A24,FUNCT_1:47;
    hence thesis by A12,A19,A22,MESFUNC1:def 3;
  end;
  f1 is nonnegative by A3,Th15;
  then
  integral+(M,f1+g1) = integral+(M,f1) + integral+(M,g1) by A10,A8,A17,A7,A16
,A14,A6,Lm10;
  hence thesis by A10,A8,A12,A19,A21,FUNCT_1:2;
end;
