theorem :: MESFUN17:66

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))) | [:I,J:]),x) holds
( P1Gz is continuous & dom (ProjPMap1 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),x)) = J & P1Gz | J is bounded & P1Gz is_integrable_on J & integral (P1Gz,J) = Integral (L-Meas,(ProjPMap1 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),x))) & integral (P1Gz,J) = (Integral2 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) . x & ProjPMap1 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),x) is_integrable_on L-Meas )