theorem Th44: :: MESFUN16:44

for I being Subset of REAL
for J being non empty closed_interval Subset of REAL
for x being Element of REAL
for f being PartFunc of [:RNS_Real,RNS_Real:],RNS_Real
for g being PartFunc of [: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 )