theorem Th43: :: MESFUN16:43

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 is_integrable_on L-Meas & integral (Pg2,I) = Integral (L-Meas,Pg2) & integral (Pg2,I) = Integral (L-Meas,(ProjPMap2 ((R_EAL g),y))) & integral (Pg2,I) = (Integral1 (L-Meas,(R_EAL g))) . y )