
theorem Th89:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M1 be sigma_Measure of S1, B be Element of S1
 st M1.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 X2,ExtREAL st
          (for y be Element of X2 holds
             F.y = M1.(Measurable-Y-section(E,y) /\ B))
        & (for V be Element of S2 holds F is V-measurable))}
proof
   let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
   M1 be sigma_Measure of S1, B be Element of S1;
   set K = {E where E is Element of sigma measurable_rectangles(S1,S2) :
    (ex F be Function of X2,ExtREAL st
         (for y be Element of X2 holds
               F.y = M1.(Measurable-Y-section(E,y) /\ B))
       & (for V be Element of S2 holds F is V-measurable))};
   assume M1.B < +infty; then
A1:K is MonotoneClass of [:X1,X2:] by Th85;
A2:Field_generated_by measurable_rectangles(S1,S2) c= K by Th81;
   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;
