
theorem Th114:
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:]
  & product_sigma_Measure(M1,M2).V < +infty & M2.B < +infty holds
   sigma 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;
   set K = {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)};
   assume that
A1:M2 is sigma_finite and
A2:V = [:A,B:] and
A3:product_sigma_Measure(M1,M2).V < +infty and
A4:M2.B < +infty;
A5:K is MonotoneClass of [:X1,X2:] by A1,A2,A3,A4,Th112;
A6:Field_generated_by measurable_rectangles(S1,S2) c= K by A1,A2,Th106;
   sigma Field_generated_by measurable_rectangles(S1,S2)
    = sigma DisUnion measurable_rectangles(S1,S2) by SRINGS_3:22
   .= sigma measurable_rectangles(S1,S2) by Th1;
   hence thesis by A5,A6,Th87;
end;
