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 Pg2 being PartFunc of (CarProduct ((Seg n) --> 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 & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) holds
( Pg2 is_integrable_on Prod_Measure (L-Meas n) & Integral ((Prod_Measure (L-Meas n)),(ProjPMap2 (|.(R_EAL g).|,y))) = (Integral1 ((Prod_Measure (L-Meas n)),|.(R_EAL g).|)) . y )
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 Pg2 being PartFunc of (CarProduct ((Seg n) --> 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 & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) holds
( Pg2 is_integrable_on Prod_Measure (L-Meas n) & Integral ((Prod_Measure (L-Meas n)),(ProjPMap2 (|.(R_EAL g).|,y))) = (Integral1 ((Prod_Measure (L-Meas n)),|.(R_EAL g).|)) . y )
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 Pg2 being PartFunc of (CarProduct ((Seg n) --> 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 & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) holds
( Pg2 is_integrable_on Prod_Measure (L-Meas n) & Integral ((Prod_Measure (L-Meas n)),(ProjPMap2 (|.(R_EAL g).|,y))) = (Integral1 ((Prod_Measure (L-Meas n)),|.(R_EAL g).|)) . y )
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 Pg2 being PartFunc of (CarProduct ((Seg n) --> 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 & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) holds
( Pg2 is_integrable_on Prod_Measure (L-Meas n) & Integral ((Prod_Measure (L-Meas n)),(ProjPMap2 (|.(R_EAL g).|,y))) = (Integral1 ((Prod_Measure (L-Meas n)),|.(R_EAL g).|)) . y )
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 Pg2 being PartFunc of (CarProduct ((Seg n) --> 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 & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) holds
( Pg2 is_integrable_on Prod_Measure (L-Meas n) & Integral ((Prod_Measure (L-Meas n)),(ProjPMap2 (|.(R_EAL g).|,y))) = (Integral1 ((Prod_Measure (L-Meas n)),|.(R_EAL g).|)) . y )
let g be PartFunc of [:(CarProduct ((Seg n) --> REAL)),REAL:],REAL; :: thesis: for Pg2 being PartFunc of (CarProduct ((Seg n) --> 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 & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) holds
( Pg2 is_integrable_on Prod_Measure (L-Meas n) & Integral ((Prod_Measure (L-Meas n)),(ProjPMap2 (|.(R_EAL g).|,y))) = (Integral1 ((Prod_Measure (L-Meas n)),|.(R_EAL g).|)) . y )
let Pg2 be PartFunc of (CarProduct ((Seg n) --> 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 & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) implies ( Pg2 is_integrable_on Prod_Measure (L-Meas n) & Integral ((Prod_Measure (L-Meas n)),(ProjPMap2 (|.(R_EAL g).|,y))) = (Integral1 ((Prod_Measure (L-Meas n)),|.(R_EAL g).|)) . y ) )
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: Pg2 = ProjPMap2 (|.(R_EAL g).|,y) ; :: thesis: ( Pg2 is_integrable_on Prod_Measure (L-Meas n) & Integral ((Prod_Measure (L-Meas n)),(ProjPMap2 (|.(R_EAL g).|,y))) = (Integral1 ((Prod_Measure (L-Meas n)),|.(R_EAL g).|)) . y )
R_EAL Pg2 = ProjPMap2 (|.(R_EAL g).|,y) by A6, MESFUNC5:def 7;
hence Pg2 is_integrable_on Prod_Measure (L-Meas n) by A1, A2, A3, A4, A5, Th46; :: thesis: Integral ((Prod_Measure (L-Meas n)),(ProjPMap2 (|.(R_EAL g).|,y))) = (Integral1 ((Prod_Measure (L-Meas n)),|.(R_EAL g).|)) . y
thus Integral ((Prod_Measure (L-Meas n)),(ProjPMap2 (|.(R_EAL g).|,y))) = (Integral1 ((Prod_Measure (L-Meas n)),|.(R_EAL g).|)) . y by MESFUN12:def 7; :: thesis: verum