
theorem Th88:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M2 be sigma_Measure of S2, B be Element of S2
 st M2.B < +infty holds
   sigma measurable_rectangles(S1,S2)
   c= {E where E is Element of sigma measurable_rectangles(S1,S2):
       (ex F be Function of X1,ExtREAL st
          (for x be Element of X1 holds
             F.x = M2.(Measurable-X-section(E,x) /\ B))
        & (for V be Element of S1 holds F is V-measurable))}
proof
   let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
   M2 be sigma_Measure of S2, B be Element of S2;
   set K = {E where E is Element of sigma measurable_rectangles(S1,S2) :
     (ex F be Function of X1,ExtREAL st
          (for x be Element of X1 holds
                F.x = M2.(Measurable-X-section(E,x) /\ B))
        & (for V be Element of S1 holds F is V-measurable))};
   assume M2.B < +infty; then
A1:K is MonotoneClass of [:X1,X2:] by Th84;
A2:Field_generated_by measurable_rectangles(S1,S2) c= K by Th80;
   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 A1,A2,Th87;
end;
