let I be non empty closed_interval Subset of REAL; :: thesis: for J being Subset of REAL
for y being Element of REAL
for f being PartFunc of [:RNS_Real,RNS_Real:],RNS_Real
for g being PartFunc of [:REAL,REAL:],REAL
for Pg2 being PartFunc of REAL,REAL
for E being Element of L-Field st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) & E = I holds
Pg2 is E -measurable
let J be Subset of REAL; :: thesis: for y being Element of REAL
for f being PartFunc of [:RNS_Real,RNS_Real:],RNS_Real
for g being PartFunc of [:REAL,REAL:],REAL
for Pg2 being PartFunc of REAL,REAL
for E being Element of L-Field st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) & E = I holds
Pg2 is E -measurable
let y be Element of REAL ; :: thesis: for f being PartFunc of [:RNS_Real,RNS_Real:],RNS_Real
for g being PartFunc of [:REAL,REAL:],REAL
for Pg2 being PartFunc of REAL,REAL
for E being Element of L-Field st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) & E = I holds
Pg2 is E -measurable
let f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real; :: thesis: for g being PartFunc of [:REAL,REAL:],REAL
for Pg2 being PartFunc of REAL,REAL
for E being Element of L-Field st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) & E = I holds
Pg2 is E -measurable
let g be PartFunc of [:REAL,REAL:],REAL; :: thesis: for Pg2 being PartFunc of REAL,REAL
for E being Element of L-Field st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) & E = I holds
Pg2 is E -measurable
let Pg2 be PartFunc of REAL,REAL; :: thesis: for E being Element of L-Field st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) & E = I holds
Pg2 is E -measurable
let E be Element of L-Field ; :: thesis: ( y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) & E = I implies Pg2 is E -measurable )
assume that
A1: y in J and
A2: dom f = [:I,J:] and
A3: f is_continuous_on [:I,J:] and
A4: f = g and
A5: Pg2 = ProjPMap2 (|.(R_EAL g).|,y) and
A6: E = I ; :: thesis: Pg2 is E -measurable
A7: dom Pg2 = I by A1, A2, A4, A5, Th28;
( Pg2 | I is bounded & Pg2 is_integrable_on I ) by A1, A2, A3, A4, A5, Th47;
hence Pg2 is E -measurable by A7, A6, MESFUN14:49; :: thesis: verum