
theorem Th80:
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,
  E be Element of sigma measurable_rectangles(S1,S2) holds
  Integral1(M1,Xchi(E,[:X1,X2:])|E) = Integral1(M1,Xchi(E,[:X1,X2:]))
& Integral2(M2,Xchi(E,[:X1,X2:])|E) = Integral2(M2,Xchi(E,[:X1,X2:]))
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,
    E be Element of sigma measurable_rectangles(S1,S2);

    now let y be Element of X2;
     set XC1 = Xchi(Measurable-Y-section(E,y),X1);
A1:  ProjPMap2(Xchi(E,[:X1,X2:])|E,y)
      = ProjPMap2(Xchi(E,[:X1,X2:]),y) | Y-section(E,y) by Th34
     .= Xchi(Y-section(E,y),X1) | Y-section(E,y) by Th35
     .= Xchi(Measurable-Y-section(E,y),X1) | Y-section(E,y) by MEASUR11:def 7
     .= XC1 | Measurable-Y-section(E,y) by MEASUR11:def 7
     .= chi(+infty,Measurable-Y-section(E,y),X1)|Measurable-Y-section(E,y)
         by Th2;

     Integral1(M1,Xchi(E,[:X1,X2:])|E).y
      = Integral(M1,ProjPMap2(Xchi(E,[:X1,X2:])|E,y)) by Def7
     .= +infty * M1.Measurable-Y-section(E,y) by A1,Th50
     .= Integral(M1,chi(+infty,Measurable-Y-section(E,y),X1)) by Th49
     .= Integral( M1,Xchi(Measurable-Y-section(E,y),X1) ) by Th2
     .= Integral( M1,Xchi(Y-section(E,y),X1) ) by MEASUR11:def 7
     .= Integral(M1,ProjPMap2(Xchi(E,[:X1,X2:]),y) ) by Th35;
     hence Integral1(M1,Xchi(E,[:X1,X2:])|E).y
      = Integral1(M1,Xchi(E,[:X1,X2:])).y by Def7;
    end;
    hence Integral1(M1,Xchi(E,[:X1,X2:])|E)
     = Integral1(M1,Xchi(E,[:X1,X2:])) by FUNCT_2:def 8;

    now let x be Element of X1;
     set XC2 = Xchi(Measurable-X-section(E,x),X2);
A1:  ProjPMap1(Xchi(E,[:X1,X2:])|E,x)
      = ProjPMap1(Xchi(E,[:X1,X2:]),x) | X-section(E,x) by Th34
     .= Xchi(X-section(E,x),X2) | X-section(E,x) by Th35
     .= Xchi(Measurable-X-section(E,x),X2) | X-section(E,x) by MEASUR11:def 6
     .= XC2 | Measurable-X-section(E,x) by MEASUR11:def 6
     .= chi(+infty,Measurable-X-section(E,x),X2)|Measurable-X-section(E,x)
         by Th2;

     Integral2(M2,Xchi(E,[:X1,X2:])|E).x
      = Integral(M2,ProjPMap1(Xchi(E,[:X1,X2:])|E,x)) by Def8
     .= +infty * M2.Measurable-X-section(E,x) by A1,Th50
     .= Integral(M2,chi(+infty,Measurable-X-section(E,x),X2)) by Th49
     .= Integral( M2,Xchi(Measurable-X-section(E,x),X2) ) by Th2
     .= Integral( M2,Xchi(X-section(E,x),X2) ) by MEASUR11:def 6
     .= Integral(M2,ProjPMap1(Xchi(E,[:X1,X2:]),x) ) by Th35;
     hence Integral2(M2,Xchi(E,[:X1,X2:])|E).x
      = Integral2(M2,Xchi(E,[:X1,X2:])).x by Def8;
    end;
    hence Integral2(M2,Xchi(E,[:X1,X2:])|E)
     = Integral2(M2,Xchi(E,[:X1,X2:])) by FUNCT_2:def 8;
end;
