let n be non zero Nat; :: thesis: for I being Subset of (CarProduct ((Seg n) --> REAL))
for J being 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 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
let I be Subset of (CarProduct ((Seg n) --> REAL)); :: thesis: for J being 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 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
let J be closed_interval Subset of REAL; :: thesis: 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 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
let x be Element of CarProduct ((Seg n) --> REAL); :: thesis: 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 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
let f be PartFunc of [:(CarProduct ((Seg n) --> RNS_Real)),RNS_Real:],RNS_Real; :: thesis: for g being PartFunc of [:(CarProduct ((Seg n) --> REAL)),REAL:],REAL
for Pg1 being PartFunc of REAL,REAL st 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
let g be PartFunc of [:(CarProduct ((Seg n) --> REAL)),REAL:],REAL; :: thesis: for Pg1 being PartFunc of REAL,REAL st 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
let Pg1 be PartFunc of REAL,REAL; :: thesis: ( dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) implies Pg1 is_integrable_on L-Meas )
assume that
A1: dom f = [:I,J:] and
A2: f is_continuous_on [:I,J:] and
A3: f = g and
A4: Pg1 = ProjPMap1 (|.(R_EAL g).|,x) ; :: thesis: Pg1 is_integrable_on L-Meas

per cases ( x in I or not x in I ) ;

suppose x in I ; :: thesis: Pg1 is_integrable_on L-Meas

then A5: dom Pg1 = J by A1, A3, A4, MESFUN16:27;
A6: J is Element of L-Field by MEASUR10:5, MEASUR12:75;

now :: thesis: ( J <> {} implies Pg1 is_integrable_on L-Meas )

assume J <> {} ; :: thesis: Pg1 is_integrable_on L-Meas
then reconsider J = J as non empty closed_interval Subset of REAL ;
( Pg1 | J is bounded & Pg1 is_integrable_on J ) by A1, A2, A3, A4, A5, Th25, INTEGRA5:10, INTEGRA5:11;
hence Pg1 is_integrable_on L-Meas by A5, A6, MESFUN14:49; :: thesis: verum

end;

hence Pg1 is_integrable_on L-Meas by A5, Th1; :: thesis: verum

end;

suppose not x in I ; :: thesis: Pg1 is_integrable_on L-Meas

then dom Pg1 = {} by A1, A3, A4, MESFUN16:27;
hence Pg1 is_integrable_on L-Meas by Th1; :: thesis: verum

end;

end;