
theorem Th79:
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,chi(E,[:X1,X2:])|E) = Integral1(M1,chi(E,[:X1,X2:]))
& Integral2(M2,chi(E,[:X1,X2:])|E) = Integral2(M2,chi(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;
A1:  ProjPMap2(chi(E,[:X1,X2:])|E,y)
      = ProjPMap2(chi(E,[:X1,X2:]),y) | Y-section(E,y) by Th34
     .= chi(Y-section(E,y),X1) | Y-section(E,y) by Th48
     .= chi(Measurable-Y-section(E,y),X1) | Y-section(E,y) by MEASUR11:def 7
     .= chi(Measurable-Y-section(E,y),X1) | Measurable-Y-section(E,y)
       by MEASUR11:def 7;

     Integral1(M1,chi(E,[:X1,X2:])|E).y
      = Integral(M1,ProjPMap2(chi(E,[:X1,X2:])|E,y)) by Def7
     .= M1.Measurable-Y-section(E,y) by A1,MESFUNC9:14
     .= Integral( M1,chi(Measurable-Y-section(E,y),X1) )
          by MESFUNC9:14
     .= Integral( M1,chi(Y-section(E,y),X1) ) by MEASUR11:def 7
     .= Integral(M1,ProjPMap2(chi(E,[:X1,X2:]),y) ) by Th48;
     hence Integral1(M1,chi(E,[:X1,X2:])|E).y
      = Integral1(M1,chi(E,[:X1,X2:])).y by Def7;
    end;
    hence Integral1(M1,chi(E,[:X1,X2:])|E) = Integral1(M1,chi(E,[:X1,X2:]))
      by FUNCT_2:def 8;

    now let x be Element of X1;
A2:  ProjPMap1(chi(E,[:X1,X2:])|E,x)
      = ProjPMap1(chi(E,[:X1,X2:]),x) | X-section(E,x) by Th34
     .= chi(X-section(E,x),X2) | X-section(E,x) by Th48
     .= chi(Measurable-X-section(E,x),X2) | X-section(E,x) by MEASUR11:def 6
     .= chi(Measurable-X-section(E,x),X2) | Measurable-X-section(E,x)
       by MEASUR11:def 6;

     Integral2(M2,chi(E,[:X1,X2:])|E).x
      = Integral(M2,ProjPMap1(chi(E,[:X1,X2:])|E,x)) by Def8
     .= M2.Measurable-X-section(E,x) by A2,MESFUNC9:14
     .= Integral( M2,chi(Measurable-X-section(E,x),X2) )
          by MESFUNC9:14
     .= Integral( M2,chi(X-section(E,x),X2) ) by MEASUR11:def 6
     .= Integral(M2,ProjPMap1(chi(E,[:X1,X2:]),x) ) by Th48;
     hence Integral2(M2,chi(E,[:X1,X2:])|E).x
      = Integral2(M2,chi(E,[:X1,X2:])).x by Def8;
    end;
    hence Integral2(M2,chi(E,[:X1,X2:])|E) = Integral2(M2,chi(E,[:X1,X2:]))
      by FUNCT_2:def 8;
end;
