theorem Th53: :: MESFUN12:53

for X1, X2 being non empty set
for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite holds
( (X-vol (E,M1)) . y = Integral (M1,(ProjPMap2 ((chi (E,[:X1,X2:])),y))) & (X-vol (E,M1)) . y = integral+ (M1,(ProjPMap2 ((chi (E,[:X1,X2:])),y))) & (X-vol (E,M1)) . y = integral' (M1,(ProjPMap2 ((chi (E,[:X1,X2:])),y))) )