theorem Th59: :: MESFUN17:59

for x being Element of REAL
for I, J, K being non empty closed_interval Subset 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 P1Gz being PartFunc of REAL,REAL st x in I & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & P1Gz = (ProjPMap1 ((Integral2 (L-Meas,(R_EAL g))),x)) | J holds
( P1Gz || J is bounded & P1Gz is_integrable_on J )