theorem Th32: :: MESFUN18:32

for n being non zero Nat
for I being Subset of (CarProduct ((Seg n) --> REAL))
for J being non empty closed_interval Subset of REAL
for x being Element of CarProduct ((Seg n) --> REAL)
for f being PartFunc of [:(CarProduct ((Seg n) --> RNS_Real)),RNS_Real:],RNS_Real
for g being PartFunc of [:(CarProduct ((Seg n) --> 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
( 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 )