
theorem Th78:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M2 be sigma_Measure of S2,
  E be Element of sigma measurable_rectangles(S1,S2)
st
  E in Field_generated_by measurable_rectangles(S1,S2)
holds
  (for B be Element of S2 holds
     E in {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,
   E be Element of sigma measurable_rectangles(S1,S2);
   assume A0: E in Field_generated_by measurable_rectangles(S1,S2);
   let B be Element of 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)) by A0,Th76;
    hence thesis;
end;
