theorem Th48: :: MESFUN18:48

for n being non zero Nat
for D being b₁ -element FinSequence
for J being Subset of REAL
for y being Element of REAL
for f being PartFunc of [:(CarProduct ((Seg n) --> RNS_Real)),RNS_Real:],RNS_Real
for g being PartFunc of [:(CarProduct ((Seg n) --> REAL)),REAL:],REAL
for Pg2 being PartFunc of (CarProduct ((Seg n) --> REAL)),REAL st ( for i being Nat st i in Seg n holds
D . i is closed_interval Subset of REAL ) & y in J & dom f = [:(CarProduct D),J:] & f is_continuous_on [:(CarProduct D),J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) holds
( Pg2 is_integrable_on Prod_Measure (L-Meas n) & Integral ((Prod_Measure (L-Meas n)),(ProjPMap2 (|.(R_EAL g).|,y))) = (Integral1 ((Prod_Measure (L-Meas n)),|.(R_EAL g).|)) . y )