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