
theorem Th81:
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 holds
Field_generated_by 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;
   now let E be set;
     assume A1: 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 A1;
     E1 in Field_generated_by measurable_rectangles(S1,S2) by A1;
     hence E in
   {E where E is Element of sigma measurable_rectangles(S1,S2) :
     (ex F be Function of X2,ExtREAL st
          (for x be Element of X2 holds
                F.x = M1.(Measurable-Y-section(E,x) /\ B))
        & (for V be Element of S2 holds F is V-measurable))}by Th79;
   end;
   hence thesis;
end;
