theorem :: MESFUN17:67

for y 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 P2Gz being PartFunc of REAL,REAL st y in J & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & P2Gz = ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) holds
( P2Gz is continuous & dom (ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y)) = I & P2Gz | I is bounded & P2Gz is_integrable_on I & integral (P2Gz,I) = Integral (L-Meas,(ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y))) & integral (P2Gz,I) = (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) . y & ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) is_integrable_on L-Meas )