theorem Th54: :: MESFUN18:54

for n being non zero Nat
for D being b₁ -element FinSequence
for J being closed_interval Subset 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 G1 being PartFunc of REAL,REAL st ( for i being Nat st i in Seg n holds
D . i is closed_interval Subset of REAL ) & [:(CarProduct D),J:] = dom f & f is_continuous_on [:(CarProduct D),J:] & f = g & G1 = (Integral1 ((Prod_Measure (L-Meas n)),(R_EAL g))) | J holds
G1 is continuous