theorem Th24: :: MESFUN17:24

for I, J being Subset of REAL
for K being non empty closed_interval Subset of REAL
for x, y being Element of REAL
for f being PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real
for g being PartFunc of [:[:REAL,REAL:],REAL:],REAL
for Pg1 being PartFunc of REAL,REAL st x in I & y in J & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) holds
( Pg1 | K is bounded & Pg1 is_integrable_on K )