
theorem Th59:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M1 be sigma_Measure of S1, f be PartFunc of [:X1,X2:],ExtREAL,
  E be Element of sigma measurable_rectangles(S1,S2), V be Element of S2
 st M1 is sigma_finite & (f is nonnegative or f is nonpositive)
  & E = dom f & f is E-measurable
 holds
  Integral1(M1,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, f be PartFunc of [:X1,X2:],ExtREAL,
    A be Element of sigma measurable_rectangles(S1,S2), V be Element of S2;
    assume that
A1:  M1 is sigma_finite and
A3:  f is nonnegative or f is nonpositive and
A4:  A = dom f and
A5:  f is A-measurable;
    consider I1 be Function of X2,ExtREAL such that
A6:  for y be Element of X2 holds I1.y = Integral(M1,ProjPMap2(f,y)) and
A7:  for W be Element of S2 holds I1 is W-measurable
       by A1,A3,A4,A5,Lm7,Lm8;
    I1 = Integral1(M1,f) by A6,Def7;
    hence Integral1(M1,f) is V-measurable by A7;
end;
