
theorem Th74:
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 nonnegative & f is E1-measurable
   & E2 = dom g & g is nonnegative & 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 nonnegative and
A3:  f is A-measurable and
A4:  B = dom g and
A5:  g is nonnegative and
A6:  g is B-measurable;
A7: dom(f+g) = A /\ B by A1,A2,A4,A5,MESFUNC5:22;
    set f1 = f|(A/\B), g1 = g|(A/\B);
A8: dom f1 = A /\ B & dom g1 = A /\ B by A1,A4,XBOOLE_1:17,RELAT_1:62;
A9: dom f /\ (A /\ B) = A /\ B & dom g /\ (A /\ B) = A /\ B
      by A1,A4,XBOOLE_1:17,28;
A10:f is (A/\B)-measurable & g is (A/\B)-measurable
      by A3,A6,XBOOLE_1:17,MESFUNC1:30; then
A11:f1 is (A/\B)-measurable & g1 is (A/\B)-measurable
      by A9,MESFUNC5:42;
A12:f1 is nonnegative & g1 is nonnegative by A2,A5,MESFUNC5:15; then
A13:Integral1(M1,f1) is nonnegative & Integral1(M1,g1) is nonnegative
  & Integral2(M2,f1) is nonnegative & Integral2(M2,g1) is nonnegative
      by A8,A11,Th66; then
    reconsider IF1 = Integral1(M1,f1), IG1 = Integral1(M1,g1)
      as without-infty Function of X2,ExtREAL;
    reconsider IF2 = Integral2(M2,f1), IG2 = Integral2(M2,g1)
      as without-infty Function of X1,ExtREAL by A13;
A14:IF1+IG1 = Integral1(M1,f1) + Integral1(M1,g1)
  & IF2+IG2 = Integral2(M2,f1) + Integral2(M2,g1);

A21:f+g is nonnegative by A2,A5,MESFUNC5:22;

    for y be Element of X2 holds
     (Integral1(M1,f1)+Integral1(M1,g1)).y = Integral1(M1,f+g).y
    proof
     let y be Element of X2;
     dom(ProjPMap2(f1,y)) = Y-section(A /\ B,y)
   & dom(ProjPMap2(g1,y)) = Y-section(A /\ B,y) by A8,Def4; then
A15: dom(ProjPMap2(f1,y)) = Measurable-Y-section(A /\ B,y)
   & dom(ProjPMap2(g1,y)) = Measurable-Y-section(A /\ B,y) by MEASUR11:def 7;
     ProjPMap2(f1,y) is (Measurable-Y-section(A/\B,y))-measurable
   & ProjPMap2(g1,y) is (Measurable-Y-section(A/\B,y))-measurable
        by A8,A11,Th47; then
A16: Integral(M1,ProjPMap2(f1,y)) = integral+(M1,ProjPMap2(f1,y))
   & Integral(M1,ProjPMap2(g1,y)) = integral+(M1,ProjPMap2(g1,y))
       by A12,A15,Th32,MESFUNC5:88;

A17: ProjPMap2(f+g,y) = ProjPMap2(f,y) + ProjPMap2(g,y) by Th44;

     ProjPMap2(f1,y) = ProjPMap2(f,y)|Y-section(A /\ B,y)
   & ProjPMap2(g1,y) = ProjPMap2(g,y)|Y-section(A /\ B,y)
      by Th34; then
A18: ProjPMap2(f1,y) = ProjPMap2(f,y)|Measurable-Y-section(A /\ B,y)
   & ProjPMap2(g1,y) = ProjPMap2(g,y)|Measurable-Y-section(A /\ B,y)
       by MEASUR11:def 7;

     dom(ProjPMap2(f,y)) = Y-section(A,y)
   & dom(ProjPMap2(g,y)) = Y-section(B,y) by A1,A4,Def4; then
A19: dom(ProjPMap2(f,y)) = Measurable-Y-section(A,y)
   & dom(ProjPMap2(g,y)) = Measurable-Y-section(B,y) by MEASUR11:def 7;

     dom(ProjPMap2(f+g,y)) = Y-section(A /\ B,y) by A7,Def4; then
A20: Measurable-Y-section(A /\ B,y) = dom(ProjPMap2(f+g,y)) by MEASUR11:def 7;

     f+g is (A/\B)-measurable by A2,A5,A10,MESFUNC5:31; then
A22: ProjPMap2(f+g,y) is (Measurable-Y-section(A/\B,y))-measurable
       by A7,Th47;

A23: (Integral1(M1,f1)+Integral1(M1,g1)).y
      = Integral1(M1,f1).y + Integral1(M1,g1).y by A13,DBLSEQ_3:7
     .= Integral(M1,ProjPMap2(f1,y)) + Integral1(M1,g1).y by Def7
     .= integral+(M1,ProjPMap2(f1,y)) + integral+(M1,ProjPMap2(g1,y))
       by A16,Def7;

     ProjPMap2(f,y) is nonnegative & ProjPMap2(g,y) is nonnegative
   & ProjPMap2(f,y) is (Measurable-Y-section(A,y))-measurable
   & ProjPMap2(g,y) is (Measurable-Y-section(B,y))-measurable
       by A1,A3,A4,A6,A2,A5,Th32,Th47; then
     ex C be Element of S1 st
      C = dom(ProjPMap2(f,y)+ProjPMap2(g,y)) &
      integral+(M1,ProjPMap2(f,y)+ProjPMap2(g,y))
       = integral+(M1,ProjPMap2(f,y)|C) + integral+(M1,ProjPMap2(g,y)|C)
          by A19,MESFUNC5:78; then
     (Integral1(M1,f1)+Integral1(M1,g1)).y
       = Integral(M1,ProjPMap2(f+g,y))
      by A17,A18,A20,A23,A21,A22,Th32,MESFUNC5:88;
     hence Integral1(M1,f+g).y
      = (Integral1(M1,f1) + Integral1(M1,g1)).y by Def7;
    end;
    hence Integral1(M1,f+g)
      = Integral1(M1,f|dom(f+g)) + Integral1(M1,g|dom(f+g))
        by A7,A14,FUNCT_2:63;
    for x be Element of X1 holds
     (Integral2(M2,f1)+Integral2(M2,g1)).x = Integral2(M2,f+g).x
    proof
     let x be Element of X1;
     dom(ProjPMap1(f1,x)) = X-section(A /\ B,x)
   & dom(ProjPMap1(g1,x)) = X-section(A /\ B,x) by A8,Def3; then
B15: dom(ProjPMap1(f1,x)) = Measurable-X-section(A /\ B,x)
   & dom(ProjPMap1(g1,x)) = Measurable-X-section(A /\ B,x) by MEASUR11:def 6;
     ProjPMap1(f1,x) is (Measurable-X-section(A/\B,x))-measurable
   & ProjPMap1(g1,x) is (Measurable-X-section(A/\B,x))-measurable
        by A8,A11,Th47; then
B16: Integral(M2,ProjPMap1(f1,x)) = integral+(M2,ProjPMap1(f1,x))
   & Integral(M2,ProjPMap1(g1,x)) = integral+(M2,ProjPMap1(g1,x))
       by A12,B15,Th32,MESFUNC5:88;

B17: ProjPMap1(f+g,x) = ProjPMap1(f,x) + ProjPMap1(g,x) by Th44;

     ProjPMap1(f1,x) = ProjPMap1(f,x)|X-section(A /\ B,x)
   & ProjPMap1(g1,x) = ProjPMap1(g,x)|X-section(A /\ B,x)
      by Th34; then
B18: ProjPMap1(f1,x) = ProjPMap1(f,x)|Measurable-X-section(A /\ B,x)
   & ProjPMap1(g1,x) = ProjPMap1(g,x)|Measurable-X-section(A /\ B,x)
       by MEASUR11:def 6;
     dom(ProjPMap1(f,x)) = X-section(A,x)
   & dom(ProjPMap1(g,x)) = X-section(B,x) by A1,A4,Def3; then
B19: dom(ProjPMap1(f,x)) = Measurable-X-section(A,x)
   & dom(ProjPMap1(g,x)) = Measurable-X-section(B,x) by MEASUR11:def 6;
     dom(ProjPMap1(f+g,x)) = X-section(A /\ B,x) by A7,Def3; then
B20: Measurable-X-section(A /\ B,x) = dom(ProjPMap1(f+g,x)) by MEASUR11:def 6;
     f+g is (A/\B)-measurable by A2,A5,A10,MESFUNC5:31; then
B22: ProjPMap1(f+g,x) is (Measurable-X-section(A/\B,x))-measurable
       by A7,Th47;
B23: (Integral2(M2,f1)+Integral2(M2,g1)).x
      = Integral2(M2,f1).x + Integral2(M2,g1).x by A13,DBLSEQ_3:7
     .= Integral(M2,ProjPMap1(f1,x)) + Integral2(M2,g1).x by Def8
     .= integral+(M2,ProjPMap1(f1,x)) + integral+(M2,ProjPMap1(g1,x))
       by B16,Def8;
     ProjPMap1(f,x) is nonnegative & ProjPMap1(g,x) is nonnegative
   & ProjPMap1(f,x) is (Measurable-X-section(A,x))-measurable
   & ProjPMap1(g,x) is (Measurable-X-section(B,x))-measurable
       by A1,A3,A4,A6,A2,A5,Th32,Th47; then
     ex C be Element of S2 st
      C = dom(ProjPMap1(f,x)+ProjPMap1(g,x)) &
      integral+(M2,ProjPMap1(f,x)+ProjPMap1(g,x))
       = integral+(M2,ProjPMap1(f,x)|C) + integral+(M2,ProjPMap1(g,x)|C)
          by B19,MESFUNC5:78; then
     (Integral2(M2,f1)+Integral2(M2,g1)).x
       = Integral(M2,ProjPMap1(f+g,x))
      by B17,B18,B20,B23,A21,B22,Th32,MESFUNC5:88;
     hence Integral2(M2,f+g).x
      = (Integral2(M2,f1) + Integral2(M2,g1)).x by Def8;
    end;
    hence Integral2(M2,f+g)
      = Integral2(M2,f|dom(f+g)) + Integral2(M2,g|dom(f+g))
        by A7,A14,FUNCT_2:63;
end;
