theorem :: MESFUN12:38

for X1, X2 being non empty set
for S1 being SigmaField of X1
for S2 being SigmaField of X2
for E being Element of sigma (measurable_rectangles (S1,S2))
for M2 being sigma_Measure of S2
for A being Element of S1
for B being Element of S2
for x being Element of X1 holds (M2 . (B /\ (Measurable-X-section (E,x)))) * ((chi (A,X1)) . x) = Integral (M2,(ProjPMap1 (((chi ([:A,B:],[:X1,X2:])) | E),x)))