theorem :: MESFUN12:84

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 M2 being sigma_Measure of S2
for A being Element of sigma (measurable_rectangles (S1,S2))
for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & ( f is nonnegative or f is nonpositive ) & A = dom f & f is A -measurable holds
( Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) & Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) ) by Lm16, Lm18, Lm17, Lm19;