let I be Subset of REAL; :: thesis: for J being non empty closed_interval Subset of REAL
for x being Element of REAL
for f being PartFunc of [:RNS_Real,RNS_Real:],RNS_Real
for g being PartFunc of [:REAL,REAL:],REAL
for Pg1 being PartFunc of REAL,REAL
for E being Element of L-Field st x in I & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) & E = J holds
Pg1 is E -measurable
let J be non empty closed_interval Subset of REAL; :: thesis: for x being Element of REAL
for f being PartFunc of [:RNS_Real,RNS_Real:],RNS_Real
for g being PartFunc of [:REAL,REAL:],REAL
for Pg1 being PartFunc of REAL,REAL
for E being Element of L-Field st x in I & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) & E = J holds
Pg1 is E -measurable
let x 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 Pg1 being PartFunc of REAL,REAL
for E being Element of L-Field st x in I & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) & E = J holds
Pg1 is E -measurable
let f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real; :: thesis: for g being PartFunc of [:REAL,REAL:],REAL
for Pg1 being PartFunc of REAL,REAL
for E being Element of L-Field st x in I & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) & E = J holds
Pg1 is E -measurable
let g be PartFunc of [:REAL,REAL:],REAL; :: thesis: for Pg1 being PartFunc of REAL,REAL
for E being Element of L-Field st x in I & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) & E = J holds
Pg1 is E -measurable
let Pg1 be PartFunc of REAL,REAL; :: thesis: for E being Element of L-Field st x in I & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) & E = J holds
Pg1 is E -measurable
let E be Element of L-Field ; :: thesis: ( x in I & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) & E = J implies Pg1 is E -measurable )
assume that
A1: x in I and
A2: dom f = [:I,J:] and
A3: f is_continuous_on [:I,J:] and
A4: f = g and
A5: Pg1 = ProjPMap1 (|.(R_EAL g).|,x) and
A6: E = J ; :: thesis: Pg1 is E -measurable
A7: dom Pg1 = J by A1, A2, A4, A5, Th27;
( Pg1 | J is bounded & Pg1 is_integrable_on J ) by A1, A2, A3, A4, A5, Th44;
hence Pg1 is E -measurable by A7, A6, MESFUN14:49; :: thesis: verum