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 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]) holds
( Pg1 is_integrable_on L-Meas & integral (Pg1,K) = Integral (L-Meas,Pg1) & integral (Pg1,K) = Integral (L-Meas,(ProjPMap1 (|.(R_EAL g).|,[x,y]))) & integral (Pg1,K) = (Integral2 (L-Meas,|.(R_EAL g).|)) . [x,y] )
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 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]) holds
( Pg1 is_integrable_on L-Meas & integral (Pg1,K) = Integral (L-Meas,Pg1) & integral (Pg1,K) = Integral (L-Meas,(ProjPMap1 (|.(R_EAL g).|,[x,y]))) & integral (Pg1,K) = (Integral2 (L-Meas,|.(R_EAL g).|)) . [x,y] )
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 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]) holds
( Pg1 is_integrable_on L-Meas & integral (Pg1,K) = Integral (L-Meas,Pg1) & integral (Pg1,K) = Integral (L-Meas,(ProjPMap1 (|.(R_EAL g).|,[x,y]))) & integral (Pg1,K) = (Integral2 (L-Meas,|.(R_EAL g).|)) . [x,y] )
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 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]) holds
( Pg1 is_integrable_on L-Meas & integral (Pg1,K) = Integral (L-Meas,Pg1) & integral (Pg1,K) = Integral (L-Meas,(ProjPMap1 (|.(R_EAL g).|,[x,y]))) & integral (Pg1,K) = (Integral2 (L-Meas,|.(R_EAL g).|)) . [x,y] )
let g be PartFunc of [:[:REAL,REAL:],REAL:],REAL; :: thesis: for Pg1 being PartFunc of REAL,REAL 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]) holds
( Pg1 is_integrable_on L-Meas & integral (Pg1,K) = Integral (L-Meas,Pg1) & integral (Pg1,K) = Integral (L-Meas,(ProjPMap1 (|.(R_EAL g).|,[x,y]))) & integral (Pg1,K) = (Integral2 (L-Meas,|.(R_EAL g).|)) . [x,y] )
let Pg1 be PartFunc of REAL,REAL; :: 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]) implies ( Pg1 is_integrable_on L-Meas & integral (Pg1,K) = Integral (L-Meas,Pg1) & integral (Pg1,K) = Integral (L-Meas,(ProjPMap1 (|.(R_EAL g).|,[x,y]))) & integral (Pg1,K) = (Integral2 (L-Meas,|.(R_EAL g).|)) . [x,y] ) )
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]) ; :: thesis: ( Pg1 is_integrable_on L-Meas & integral (Pg1,K) = Integral (L-Meas,Pg1) & integral (Pg1,K) = Integral (L-Meas,(ProjPMap1 (|.(R_EAL g).|,[x,y]))) & integral (Pg1,K) = (Integral2 (L-Meas,|.(R_EAL g).|)) . [x,y] )
A6: K is Element of L-Field by MEASUR10:5, MEASUR12:75;
[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 A8: ( Pg1 is_integrable_on L-Meas & integral (Pg1,K) = Integral (L-Meas,Pg1) ) by A6, A7, MESFUN14:49; :: thesis: ( integral (Pg1,K) = Integral (L-Meas,(ProjPMap1 (|.(R_EAL g).|,[x,y]))) & integral (Pg1,K) = (Integral2 (L-Meas,|.(R_EAL g).|)) . [x,y] )
R_EAL Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) by A5, MESFUNC5:def 7;
hence ( integral (Pg1,K) = Integral (L-Meas,(ProjPMap1 (|.(R_EAL g).|,[x,y]))) & integral (Pg1,K) = (Integral2 (L-Meas,|.(R_EAL g).|)) . [x,y] ) by A8, MESFUN12:def 8; :: thesis: verum