let n be non zero Nat; :: thesis: for D being n -element FinSequence
for J being Subset of REAL
for y being Element of 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 E being Element of Prod_Field (L-Field n) st ( for i being Nat st i in Seg n holds
D . i is closed_interval Subset of REAL ) & y in J & dom f = [:(CarProduct D),J:] & f is_continuous_on [:(CarProduct D),J:] & f = g & E = CarProduct D holds
ProjPMap2 (|.(R_EAL g).|,y) is E -measurable
let D be n -element FinSequence; :: thesis: for J being Subset of REAL
for y being Element of 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 E being Element of Prod_Field (L-Field n) st ( for i being Nat st i in Seg n holds
D . i is closed_interval Subset of REAL ) & y in J & dom f = [:(CarProduct D),J:] & f is_continuous_on [:(CarProduct D),J:] & f = g & E = CarProduct D holds
ProjPMap2 (|.(R_EAL g).|,y) is E -measurable
let J be Subset of REAL; :: thesis: for y being Element of 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 E being Element of Prod_Field (L-Field n) st ( for i being Nat st i in Seg n holds
D . i is closed_interval Subset of REAL ) & y in J & dom f = [:(CarProduct D),J:] & f is_continuous_on [:(CarProduct D),J:] & f = g & E = CarProduct D holds
ProjPMap2 (|.(R_EAL g).|,y) is E -measurable
let y be Element of 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 E being Element of Prod_Field (L-Field n) st ( for i being Nat st i in Seg n holds
D . i is closed_interval Subset of REAL ) & y in J & dom f = [:(CarProduct D),J:] & f is_continuous_on [:(CarProduct D),J:] & f = g & E = CarProduct D holds
ProjPMap2 (|.(R_EAL g).|,y) is E -measurable
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 E being Element of Prod_Field (L-Field n) st ( for i being Nat st i in Seg n holds
D . i is closed_interval Subset of REAL ) & y in J & dom f = [:(CarProduct D),J:] & f is_continuous_on [:(CarProduct D),J:] & f = g & E = CarProduct D holds
ProjPMap2 (|.(R_EAL g).|,y) is E -measurable
let g be PartFunc of [:(CarProduct ((Seg n) --> REAL)),REAL:],REAL; :: thesis: for E being Element of Prod_Field (L-Field n) st ( for i being Nat st i in Seg n holds
D . i is closed_interval Subset of REAL ) & y in J & dom f = [:(CarProduct D),J:] & f is_continuous_on [:(CarProduct D),J:] & f = g & E = CarProduct D holds
ProjPMap2 (|.(R_EAL g).|,y) is E -measurable
let E be Element of Prod_Field (L-Field n); :: thesis: ( ( for i being Nat st i in Seg n holds
D . i is closed_interval Subset of REAL ) & y in J & dom f = [:(CarProduct D),J:] & f is_continuous_on [:(CarProduct D),J:] & f = g & E = CarProduct D implies ProjPMap2 (|.(R_EAL g).|,y) is E -measurable )
assume that
A1: for i being Nat st i in Seg n holds
D . i is closed_interval Subset of REAL and
A2: y in J and
A3: dom f = [:(CarProduct D),J:] and
A4: f is_continuous_on [:(CarProduct D),J:] and
A5: f = g and
A6: E = CarProduct D ; :: thesis: ProjPMap2 (|.(R_EAL g).|,y) is E -measurable
A7: ( dom (ProjPMap2 (|.(R_EAL g).|,y)) = CarProduct D & ProjPMap2 (|.(R_EAL g).|,y) is_integrable_on Prod_Measure (L-Meas n) ) by A1, A2, A3, A4, A5, Th46;
then ex A being Element of Prod_Field (L-Field n) st
( A = dom (ProjPMap2 (|.(R_EAL g).|,y)) & ProjPMap2 (|.(R_EAL g).|,y) is A -measurable ) by MESFUNC5:def 17;
hence ProjPMap2 (|.(R_EAL g).|,y) is E -measurable by A6, A7; :: thesis: verum