theorem Th47: :: MESFUN16:47

for I being non empty closed_interval Subset of REAL
for J being Subset of REAL
for y being Element of REAL
for f being PartFunc of [:RNS_Real,RNS_Real:],RNS_Real
for g being PartFunc of [:REAL,REAL:],REAL
for Pg2 being PartFunc of REAL,REAL st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) holds
( Pg2 | I is bounded & Pg2 is_integrable_on I )