theorem :: MEASUR11:73

for X1, X2 being non empty set
for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M2 being sigma_Measure of S2
for f being disjoint_valued FinSequence of measurable_rectangles (S1,S2)
for x being Element of X1
for n being Nat
for En being Element of sigma (measurable_rectangles (S1,S2))
for An being Element of S1
for Bn being Element of S2 st n in dom f & f . n = En & En = [:An,Bn:] holds
Integral (M2,(ProjMap1 ((chi ((f . n),[:X1,X2:])),x))) = (M2 . (Measurable-X-section (En,x))) * ((chi (An,X1)) . x) by Th65;