theorem Th56: :: MESFUN12:56

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))
for r being Real st M2 is sigma_finite holds
( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )