theorem Th41: :: MESFUN16:41

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 is_integrable_on L-Meas & integral (Pg1,J) = Integral (L-Meas,Pg1) & integral (Pg1,J) = Integral (L-Meas,(ProjPMap1 ((R_EAL g),x))) & integral (Pg1,J) = (Integral2 (L-Meas,(R_EAL g))) . x )