
theorem
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M1 be sigma_Measure of S1, M2 be sigma_Measure of S2,
  f,g be PartFunc of [:X1,X2:],ExtREAL,
  E1,E2 be Element of sigma measurable_rectangles(S1,S2)
  st E1 = dom f & f is nonpositive & f is E1-measurable &
     E2 = dom g & g is nonpositive & g is E2-measurable holds
  Integral1(M1,f+g) = Integral1(M1,f|dom(f+g)) + Integral1(M1,g|dom(f+g))
& Integral2(M2,f+g) = Integral2(M2,f|dom(f+g)) + Integral2(M2,g|dom(f+g))
proof
    let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
    M1 be sigma_Measure of S1, M2 be sigma_Measure of S2,
    f,g be PartFunc of [:X1,X2:],ExtREAL,
    A,B be Element of sigma measurable_rectangles(S1,S2);
    assume that
A1:  A = dom f and
A2:  f is nonpositive and
A3:  f is A-measurable and
A4:  B = dom g and
A5:  g is nonpositive and
A6:  g is B-measurable;

    reconsider f1 = -f as nonnegative PartFunc of [:X1,X2:],ExtREAL by A2;
    reconsider g1 = -g as nonnegative PartFunc of [:X1,X2:],ExtREAL by A5;

A7: f1+g1 = -(f+g) by MEASUR11:64;

A8: dom f1 = A & dom g1 = B by A1,A4,MESFUNC1:def 7; then
A9: dom(f1+g1) = A /\ B by MESFUNC5:22; then
A10:dom(f+g) = A /\ B by A7,MESFUNC1:def 7; then
A11:dom(f|dom(f+g)) = A /\ B & dom(g|dom(f+g)) = A /\ B
      by A1,A4,XBOOLE_1:17,RELAT_1:62;

A12:dom f /\ (A /\ B) = A /\ B & dom g /\ (A /\ B) = A /\ B
      by A1,A4,XBOOLE_1:17,28;

A13:f1|dom(f1+g1) = -(f|dom(f+g)) & g1|dom(f1+g1) = -(g|dom(f+g))
      by A9,A10,MESFUN11:3;

A14:f is (A/\B)-measurable & g is (A/\B)-measurable
      by A3,A6,XBOOLE_1:17,MESFUNC1:30; then
    f|dom(f+g) is (A/\B)-measurable
  & g|dom(f+g) is (A/\B)-measurable by A10,A12,MESFUNC5:42; then
A15:Integral1(M1,f1|dom(f1+g1)) = -Integral1(M1,f|dom(f+g))
  & Integral1(M1,g1|dom(f1+g1)) = -Integral1(M1,g|dom(f+g))
  & Integral2(M2,f1|dom(f1+g1)) = -Integral2(M2,f|dom(f+g))
  & Integral2(M2,g1|dom(f1+g1)) = -Integral2(M2,g|dom(f+g))
      by A11,A13,Th73;

    f+g is (A/\B)-measurable by A2,A5,A10,A14,MEASUR11:65; then
A16:Integral1(M1,f1+g1) = -Integral1(M1,f+g)
  & Integral2(M2,f1+g1) = -Integral2(M2,f+g) by A7,A10,Th73;

A17:f1 is A-measurable & g1 is B-measurable
      by A1,A3,A4,A6,MEASUR11:63; then
    Integral1(M1,f1+g1)
     = Integral1(M1,f1|dom(f1+g1)) + Integral1(M1,g1|dom(f1+g1))
       by A8,Th74; then
    -Integral1(M1,f+g)
     = -(Integral1(M1,f|dom(f+g)) + Integral1(M1,g|dom(f+g)))
       by A15,A16,MEASUR11:64; then
    Integral1(M1,f+g)
      = -(-(Integral1(M1,f|dom(f+g)) + Integral1(M1,g|dom(f+g))))
       by DBLSEQ_3:2;
    hence
    Integral1(M1,f+g) = Integral1(M1,f|dom(f+g)) + Integral1(M1,g|dom(f+g))
       by DBLSEQ_3:2;
    Integral2(M2,f1+g1)
     = Integral2(M2,f1|dom(f1+g1)) + Integral2(M2,g1|dom(f1+g1))
       by A8,A17,Th74; then
    -Integral2(M2,f+g)
     = -(Integral2(M2,f|dom(f+g)) + Integral2(M2,g|dom(f+g)))
       by A15,A16,MEASUR11:64; then
    Integral2(M2,f+g)
      = -(-(Integral2(M2,f|dom(f+g)) + Integral2(M2,g|dom(f+g))))
       by DBLSEQ_3:2;
    hence
    Integral2(M2,f+g) = Integral2(M2,f|dom(f+g)) + Integral2(M2,g|dom(f+g))
       by DBLSEQ_3:2;
end;
