
theorem Th67:
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 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 holds
  Integral1(M1,f) is nonpositive & Integral1(M1,f|E2) is nonpositive
& Integral2(M2,f) is nonpositive & Integral2(M2,f|E2) is nonpositive
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 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;

A4: f|B is nonpositive by A2,MESFUN11:1;
A5: dom(f|B) = A /\ B by A1,RELAT_1:61;

A6: f is (A/\B)-measurable by A3,XBOOLE_1:17,MESFUNC1:30;
A7: dom f /\ (A/\B) = A/\B by A1,XBOOLE_1:17,28;
    f|(A/\B) = f|A /\ f|B by RELAT_1:79; then
    f|(A/\B) = f|B by A1,RELAT_1:59,XBOOLE_1:28; then
A8: f|B is (A/\B)-measurable by A6,A7,MESFUNC5:42;

    now let y be set;
     assume y in dom(Integral1(M1,f)); then
     reconsider y1=y as Element of X2;
A9:  ProjPMap2(f,y1) is (Measurable-Y-section(A,y1))-measurable
        by A1,A3,Th47;
     dom(ProjPMap2(f,y1)) = Y-section(A,y1) by A1,Def4; then
     dom(ProjPMap2(f,y1)) = Measurable-Y-section(A,y1) by MEASUR11:def 7; then
     Integral(M1,ProjPMap2(f,y1)) <= 0 by A2,A9,Th33,MESFUN11:61;
     hence Integral1(M1,f).y <= 0 by Def7;
    end;
    hence Integral1(M1,f) is nonpositive by MESFUNC5:9;

    now let y be set;
     assume y in dom(Integral1(M1,f|B)); then
     reconsider y1=y as Element of X2;

A10: ProjPMap2(f|B,y1) is (Measurable-Y-section(A/\B,y1))-measurable
        by A5,A8,Th47;
     dom(ProjPMap2(f|B,y1)) = Y-section(A/\B,y1) by A5,Def4; then
     dom(ProjPMap2(f|B,y1)) = Measurable-Y-section(A/\B,y1)
       by MEASUR11:def 7; then
     Integral(M1,ProjPMap2(f|B,y1)) <= 0 by A4,A10,Th33,MESFUN11:61;
     hence Integral1(M1,f|B).y <= 0 by Def7;
    end;
    hence Integral1(M1,f|B) is nonpositive by MESFUNC5:9;

    now let x be set;
     assume x in dom(Integral2(M2,f)); then
     reconsider x1=x as Element of X1;
A9:  ProjPMap1(f,x1) is (Measurable-X-section(A,x1))-measurable
        by A1,A3,Th47;
     dom(ProjPMap1(f,x1)) = X-section(A,x1) by A1,Def3; then
     dom(ProjPMap1(f,x1)) = Measurable-X-section(A,x1) by MEASUR11:def 6; then
     Integral(M2,ProjPMap1(f,x1)) <= 0 by A2,A9,Th33,MESFUN11:61;
     hence Integral2(M2,f).x <= 0 by Def8;
    end;
    hence Integral2(M2,f) is nonpositive by MESFUNC5:9;

    now let x be set;
     assume x in dom(Integral2(M2,f|B)); then
     reconsider x1=x as Element of X1;

A10: ProjPMap1(f|B,x1) is (Measurable-X-section(A/\B,x1))-measurable
        by A5,A8,Th47;
     dom(ProjPMap1(f|B,x1)) = X-section(A/\B,x1) by A5,Def3; then
     dom(ProjPMap1(f|B,x1)) = Measurable-X-section(A/\B,x1)
       by MEASUR11:def 6; then
     Integral(M2,ProjPMap1(f|B,x1)) <= 0 by A4,A10,Th33,MESFUN11:61;
     hence Integral2(M2,f|B).x <= 0 by Def8;
    end;
    hence Integral2(M2,f|B) is nonpositive by MESFUNC5:9;
end;
