let I, J be non empty closed_interval Subset of REAL; :: thesis: for K being Subset of REAL
for z being Element of REAL
for f being PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real
for g being PartFunc of [:[:REAL,REAL:],REAL:],REAL
for Pg2 being PartFunc of [:REAL,REAL:],REAL
for E being Element of sigma (measurable_rectangles (L-Field,L-Field)) st z in K & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,z) & E = [:I,J:] holds
Pg2 is E -measurable
let K be Subset of REAL; :: thesis: for z being Element of REAL
for f being PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real
for g being PartFunc of [:[:REAL,REAL:],REAL:],REAL
for Pg2 being PartFunc of [:REAL,REAL:],REAL
for E being Element of sigma (measurable_rectangles (L-Field,L-Field)) st z in K & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,z) & E = [:I,J:] holds
Pg2 is E -measurable
let z be Element of REAL ; :: thesis: for f being PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real
for g being PartFunc of [:[:REAL,REAL:],REAL:],REAL
for Pg2 being PartFunc of [:REAL,REAL:],REAL
for E being Element of sigma (measurable_rectangles (L-Field,L-Field)) st z in K & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,z) & E = [:I,J:] holds
Pg2 is E -measurable
let f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real; :: thesis: for g being PartFunc of [:[:REAL,REAL:],REAL:],REAL
for Pg2 being PartFunc of [:REAL,REAL:],REAL
for E being Element of sigma (measurable_rectangles (L-Field,L-Field)) st z in K & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,z) & E = [:I,J:] holds
Pg2 is E -measurable
let g be PartFunc of [:[:REAL,REAL:],REAL:],REAL; :: thesis: for Pg2 being PartFunc of [:REAL,REAL:],REAL
for E being Element of sigma (measurable_rectangles (L-Field,L-Field)) st z in K & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,z) & E = [:I,J:] holds
Pg2 is E -measurable
let Pg2 be PartFunc of [:REAL,REAL:],REAL; :: thesis: for E being Element of sigma (measurable_rectangles (L-Field,L-Field)) st z in K & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,z) & E = [:I,J:] holds
Pg2 is E -measurable
let E be Element of sigma (measurable_rectangles (L-Field,L-Field)); :: thesis: ( z in K & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,z) & E = [:I,J:] implies Pg2 is E -measurable )
assume that
A1: z in K and
A2: dom f = [:[:I,J:],K:] and
A3: f is_continuous_on [:[:I,J:],K:] and
A4: f = g and
A5: Pg2 = ProjPMap2 (|.(R_EAL g).|,z) and
A6: E = [:I,J:] ; :: thesis: Pg2 is E -measurable
reconsider Pf2 = Pg2 as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real ;
dom Pf2 = [:I,J:] by A1, A2, A4, A5, MESFUN16:28;
hence Pg2 is E -measurable by A2, A3, A4, A5, A6, Th20, MESFUN16:50; :: thesis: verum