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 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 holds
( dom (ProjPMap2 ((R_EAL g),y)) = CarProduct D & ProjPMap2 ((R_EAL g),y) is_integrable_on Prod_Measure (L-Meas n) )
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 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 holds
( dom (ProjPMap2 ((R_EAL g),y)) = CarProduct D & ProjPMap2 ((R_EAL g),y) is_integrable_on Prod_Measure (L-Meas n) )
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 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 holds
( dom (ProjPMap2 ((R_EAL g),y)) = CarProduct D & ProjPMap2 ((R_EAL g),y) is_integrable_on Prod_Measure (L-Meas n) )
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 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 holds
( dom (ProjPMap2 ((R_EAL g),y)) = CarProduct D & ProjPMap2 ((R_EAL g),y) is_integrable_on Prod_Measure (L-Meas n) )
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 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 holds
( dom (ProjPMap2 ((R_EAL g),y)) = CarProduct D & ProjPMap2 ((R_EAL g),y) is_integrable_on Prod_Measure (L-Meas n) )
let g be PartFunc of [:(CarProduct ((Seg n) --> REAL)),REAL:],REAL; :: 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 implies ( dom (ProjPMap2 ((R_EAL g),y)) = CarProduct D & ProjPMap2 ((R_EAL g),y) is_integrable_on Prod_Measure (L-Meas n) ) )
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 ; :: thesis: ( dom (ProjPMap2 ((R_EAL g),y)) = CarProduct D & ProjPMap2 ((R_EAL g),y) is_integrable_on Prod_Measure (L-Meas n) )
reconsider RPg2 = ProjPMap2 ((R_EAL g),y) as PartFunc of (CarProduct ((Seg n) --> REAL)),REAL by MESFUN16:30;
ProjPMap2 ((R_EAL g),y) is PartFunc of (CarProduct ((Seg n) --> REAL)),REAL by MESFUN16:30;
then reconsider Pg2 = ProjPMap2 ((R_EAL g),y) as PartFunc of (CarProduct ((Seg n) --> RNS_Real)),RNS_Real by MEASUR15:42;
for i being Nat st i in Seg n holds
D . i is Subset of REAL by A1;
then A6: CarProduct D is Subset of (CarProduct ((Seg n) --> REAL)) by FINSEQ_1:3, Th11;
then dom Pg2 = CarProduct D by A2, A3, A5, MESFUN16:28;
then RPg2 is_integrable_on Prod_Measure (L-Meas n) by A1, A3, A4, A5, Th24, Th43;
hence ( dom (ProjPMap2 ((R_EAL g),y)) = CarProduct D & ProjPMap2 ((R_EAL g),y) is_integrable_on Prod_Measure (L-Meas n) ) by A6, A2, A3, A5, MESFUN16:28, MESFUNC5:def 7; :: thesis: verum