
theorem Th60:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M2 be sigma_Measure of S2, f be PartFunc of [:X1,X2:],ExtREAL,
  E be Element of sigma measurable_rectangles(S1,S2), U be Element of S1
 st M2 is sigma_finite & (f is nonnegative or f is nonpositive)
  & E = dom f & f is E-measurable
 holds
  Integral2(M2,f) is U-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, f be PartFunc of [:X1,X2:],ExtREAL,
    A be Element of sigma measurable_rectangles(S1,S2), U be Element of S1;
    assume that
A1:  M2 is sigma_finite and
A3:  f is nonnegative or f is nonpositive and
A4:  A = dom f and
A5:  f is A-measurable;
    consider I2 be Function of X1,ExtREAL such that
A6:  for x be Element of X1 holds I2.x = Integral(M2,ProjPMap1(f,x)) and
A7:  for W be Element of S1 holds I2 is W-measurable
       by A1,A3,A4,A5,Lm9,Lm10;
    I2 = Integral2(M2,f) by A6,Def8;
    hence Integral2(M2,f) is U-measurable by A7;
end;
