theorem :: MESFUN13:27

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 f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds
( Integral ((Prod_Measure (M1,M2)),f) = (Integral (M2,(Integral1 (M1,(max+ f))))) - (Integral (M2,(Integral1 (M1,(max- f))))) & Integral ((Prod_Measure (M1,M2)),f) = (Integral (M1,(Integral2 (M2,(max+ f))))) - (Integral (M1,(Integral2 (M2,(max- f))))) )