theorem Th52: :: MESFUN12:52

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 x being Element of X1
for E being Element of sigma (measurable_rectangles (S1,S2)) st M2 is sigma_finite holds
( (Y-vol (E,M2)) . x = Integral (M2,(ProjPMap1 ((chi (E,[:X1,X2:])),x))) & (Y-vol (E,M2)) . x = integral+ (M2,(ProjPMap1 ((chi (E,[:X1,X2:])),x))) & (Y-vol (E,M2)) . x = integral' (M2,(ProjPMap1 ((chi (E,[:X1,X2:])),x))) )