theorem :: MESFUN18:41

for n being non zero Nat
for D being b₁ -element FinSequence
for I being Subset of (CarProduct ((Seg n) --> RNS_Real))
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 G2 being PartFunc of (CarProduct ((Seg n) --> RNS_Real)),RNS_Real st ( for i being Nat st i in Seg n holds
D . i is closed_interval Subset of REAL ) & I = CarProduct D & [:I,J:] = dom f & f is_continuous_on [:I,J:] & f = g & G2 = (Integral2 (L-Meas,(R_EAL g))) | I holds
G2 is_continuous_on I