
theorem Th106:
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,
  V be Element of sigma measurable_rectangles(S1,S2),
  A be Element of S1, B be Element of S2
 st M2 is sigma_finite & V = [:A,B:] holds
 Field_generated_by measurable_rectangles(S1,S2)
  c= {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,
   V be Element of sigma measurable_rectangles(S1,S2),
   A be Element of S1, B be Element of S2;
   assume A1: M2 is sigma_finite & V = [:A,B:];
    let E be object;
    assume A2: E in Field_generated_by measurable_rectangles(S1,S2);
    sigma measurable_rectangles(S1,S2)
     = sigma DisUnion measurable_rectangles(S1,S2) by Th1
    .= sigma Field_generated_by measurable_rectangles(S1,S2)
      by SRINGS_3:22; then
    Field_generated_by measurable_rectangles(S1,S2)
     c= sigma measurable_rectangles(S1,S2)
       by PROB_1:def 9; then
    reconsider E1=E as Element of sigma measurable_rectangles(S1,S2) by A2;
    E1 in Field_generated_by measurable_rectangles(S1,S2) by A2;
    hence thesis by A1,Th104;
end;
