theorem :: MESFUN12:39

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 M1 being sigma_Measure of S1
for A being Element of S1
for B being Element of S2
for y being Element of X2 holds (M1 . (A /\ (Measurable-Y-section (E,y)))) * ((chi (B,X2)) . y) = Integral (M1,(ProjPMap2 (((chi ([:A,B:],[:X1,X2:])) | E),y)))