theorem Th28: :: MESFUN18:28

for n being non zero Nat
for I being Subset of (CarProduct ((Seg n) --> REAL))
for J being non empty closed_interval Subset of REAL
for x being Element of CarProduct ((Seg n) --> 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 Pg1 being PartFunc of REAL,REAL st x in I & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) holds
( Pg1 | J is bounded & Pg1 is_integrable_on J )