
theorem Th21:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 E1,E2 be Element of S, f,g be PartFunc of X,ExtREAL st
 E1 = dom f & f is nonnegative & f is E1-measurable &
 E2 = dom g & g is nonnegative & g is E2-measurable
 holds
  Integral(M,f+g) = Integral(M,f|dom(f+g)) + Integral(M,g|dom(f+g))
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    A,B be Element of S, f,g be PartFunc of X,ExtREAL;
    assume that
A1:  A = dom f and
A2:  f is nonnegative and
A3:  f is A-measurable and
A4:  B = dom g and
A5:  g is nonnegative and
A6:  g is B-measurable;

    set f1 = f|(A/\B), g1 = g|(A/\B);

A7: dom(f+g) = A /\ B by A1,A2,A4,A5,MESFUNC5:22;

A8: dom f1 = A /\ B & dom g1 = A /\ B
  & dom f /\ (A /\ B) = A /\ B & dom g /\ (A /\ B) = A /\ B
      by A1,A4,XBOOLE_1:17,28,RELAT_1:62;

A9: f is (A/\B)-measurable & g is (A/\B)-measurable
      by A3,A6,XBOOLE_1:17,MESFUNC1:30;

A10:f+g is nonnegative by A2,A5,MESFUNC5:22;

    f1 is nonnegative & g1 is nonnegative by A2,A5,MESFUNC5:15; then
A11:Integral(M,f1) = integral+(M,f1)
  & Integral(M,g1) = integral+(M,g1) by A8,A9,MESFUNC5:42,88;
    ex C be Element of S st C = dom(f+g)
   & integral+(M,f+g) = integral+(M,f|C) + integral+(M,g|C)
      by A1,A2,A3,A4,A5,A6,MESFUNC5:78;
    hence thesis by A2,A5,A7,A9,A10,A11,MESFUNC5:31,88;
end;
