
theorem Th104:
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)
st
  E in Field_generated_by measurable_rectangles(S1,S2) & M2 is sigma_finite
holds
   for V be Element of sigma measurable_rectangles(S1,S2),
       A be Element of S1, B be Element of S2 st
    V = [:A,B:] holds
    E in {E where E is Element of sigma measurable_rectangles(S1,S2) :
     Integral(M1,(Y-vol(E/\V,M2))) = (product_sigma_Measure(M1,M2)).(E/\V)}
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);
   assume that
A1: E in Field_generated_by measurable_rectangles(S1,S2) and
A2: M2 is sigma_finite;
    let V be Element of sigma measurable_rectangles(S1,S2),
    A be Element of S1, B be Element of S2;
    assume A3: V = [:A,B:];
    V in the set of all [:A,B:]
        where A is Element of S1, B is Element of S2 by A3; then
A5: V in measurable_rectangles(S1,S2) by MEASUR10:def 5;
    measurable_rectangles(S1,S2)
     c= Field_generated_by measurable_rectangles(S1,S2) by SRINGS_3:21; then
A6: E /\ V in Field_generated_by measurable_rectangles(S1,S2)
       by A1,A5,FINSUB_1:def 2;
    reconsider XX1 = X1 as Element of S1 by MEASURE1:7;
     E/\V in DisUnion measurable_rectangles(S1,S2) by A6,SRINGS_3:22; then
     E/\V in { W where W is Subset of [:X1,X2:] :
       ex G be disjoint_valued FinSequence of measurable_rectangles(S1,S2) st
         W = Union G} by SRINGS_3:def 3; then
     consider W be Subset of [:X1,X2:] such that
A11:  E/\V = W
    & ex G be disjoint_valued FinSequence of measurable_rectangles(S1,S2) st
       W = Union G;
     consider G be disjoint_valued FinSequence of measurable_rectangles(S1,S2)
      such that
A12:  E/\V = Union G by A11;
A13: G in (measurable_rectangles(S1,S2))* by FINSEQ_1:def 11;
     measurable_rectangles(S1,S2) c= sigma measurable_rectangles(S1,S2)
       by PROB_1:def 9; then
     (measurable_rectangles(S1,S2))* c= (sigma measurable_rectangles(S1,S2))*
       by FINSEQ_1:62; then
     reconsider G as disjoint_valued FinSequence
       of sigma measurable_rectangles(S1,S2) by A13,FINSEQ_1:def 11;
     Integral(M1,Y-vol(Union G,M2)) = product_sigma_Measure(M1,M2).(Union G)
       by A2,Th102;
     hence E in {E where E is Element of sigma measurable_rectangles(S1,S2) :
     Integral(M1,(Y-vol(E/\V,M2))) = (product_sigma_Measure(M1,M2)).(E/\V)}
       by A12;
end;
