theorem Th78: :: MESFUN12:78

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 E being Element of sigma (measurable_rectangles (S1,S2))
for f being PartFunc of [:X1,X2:],ExtREAL
for r being Real st E = dom f & ( f is nonnegative or f is nonpositive ) & f is E -measurable holds
( Integral1 (M1,(r (#) f)) = r (#) (Integral1 (M1,f)) & Integral2 (M2,(r (#) f)) = r (#) (Integral2 (M2,f)) )