theorem Th25: :: MESFUN18:25

for n being non zero Nat
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
for x being Element of CarProduct ((Seg n) --> REAL) st f is_continuous_on dom f & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) holds
Pg1 is continuous