let I, J be Subset of REAL; :: thesis: for K being non empty closed_interval Subset of REAL
for x, y 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 Pg1 being PartFunc of REAL,REAL
for E being Element of L-Field st x in I & y in J & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) & E = K holds
Pg1 is E -measurable
let K be non empty closed_interval Subset of REAL; :: thesis: for x, y 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 Pg1 being PartFunc of REAL,REAL
for E being Element of L-Field st x in I & y in J & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) & E = K holds
Pg1 is E -measurable
let x, y 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 Pg1 being PartFunc of REAL,REAL
for E being Element of L-Field st x in I & y in J & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) & E = K holds
Pg1 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 Pg1 being PartFunc of REAL,REAL
for E being Element of L-Field st x in I & y in J & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) & E = K holds
Pg1 is E -measurable
let g be PartFunc of [:[:REAL,REAL:],REAL:],REAL; :: thesis: for Pg1 being PartFunc of REAL,REAL
for E being Element of L-Field st x in I & y in J & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) & E = K holds
Pg1 is E -measurable
let Pg1 be PartFunc of REAL,REAL; :: thesis: for E being Element of L-Field st x in I & y in J & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) & E = K holds
Pg1 is E -measurable
let E be Element of L-Field ; :: thesis: ( x in I & y in J & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:] & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) & E = K implies Pg1 is E -measurable )
assume that
A1: ( x in I & y in J ) and
A2: dom f = [:[:I,J:],K:] and
A3: f is_continuous_on [:[:I,J:],K:] and
A4: f = g and
A5: Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) and
A6: E = K ; :: thesis: Pg1 is E -measurable
[x,y] in [:I,J:] by A1, ZFMISC_1:87;
then A7: dom Pg1 = K by A2, A4, A5, MESFUN16:27;
( Pg1 | K is bounded & Pg1 is_integrable_on K ) by A1, A2, A3, A4, A5, Th24;
hence Pg1 is E -measurable by A6, A7, MESFUN14:49; :: thesis: verum