
theorem
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
 M1 be sigma_Measure of S1, M2 be sigma_Measure of S2,
 E,E1,E2 be Element of sigma measurable_rectangles(S1,S2),
 f be PartFunc of [:X1,X2:],ExtREAL st
  E = dom f & (f is nonnegative or f is nonpositive)
& f is E-measurable & E1 misses E2
holds
  Integral1(M1,f|(E1\/E2)) = Integral1(M1,f|E1) + Integral1(M1,f|E2)
& Integral2(M2,f|(E1\/E2)) = Integral2(M2,f|E1) + Integral2(M2,f|E2)
by Lm11,Lm12;
