let I be non empty closed_interval Subset of REAL; :: thesis: for J being Subset of REAL
for y being Element of REAL
for f being PartFunc of [:RNS_Real,RNS_Real:],RNS_Real
for g being PartFunc of [:REAL,REAL:],REAL
for Pg2 being PartFunc of REAL,REAL st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) holds
( Pg2 is_integrable_on L-Meas & integral (Pg2,I) = Integral (L-Meas,Pg2) & integral (Pg2,I) = Integral (L-Meas,(ProjPMap2 (|.(R_EAL g).|,y))) & integral (Pg2,I) = (Integral1 (L-Meas,|.(R_EAL g).|)) . y )
let J be Subset of REAL; :: thesis: for y being Element of REAL
for f being PartFunc of [:RNS_Real,RNS_Real:],RNS_Real
for g being PartFunc of [:REAL,REAL:],REAL
for Pg2 being PartFunc of REAL,REAL st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) holds
( Pg2 is_integrable_on L-Meas & integral (Pg2,I) = Integral (L-Meas,Pg2) & integral (Pg2,I) = Integral (L-Meas,(ProjPMap2 (|.(R_EAL g).|,y))) & integral (Pg2,I) = (Integral1 (L-Meas,|.(R_EAL g).|)) . y )
let y 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 Pg2 being PartFunc of REAL,REAL st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) holds
( Pg2 is_integrable_on L-Meas & integral (Pg2,I) = Integral (L-Meas,Pg2) & integral (Pg2,I) = Integral (L-Meas,(ProjPMap2 (|.(R_EAL g).|,y))) & integral (Pg2,I) = (Integral1 (L-Meas,|.(R_EAL g).|)) . y )
let f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real; :: thesis: for g being PartFunc of [:REAL,REAL:],REAL
for Pg2 being PartFunc of REAL,REAL st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) holds
( Pg2 is_integrable_on L-Meas & integral (Pg2,I) = Integral (L-Meas,Pg2) & integral (Pg2,I) = Integral (L-Meas,(ProjPMap2 (|.(R_EAL g).|,y))) & integral (Pg2,I) = (Integral1 (L-Meas,|.(R_EAL g).|)) . y )
let g be PartFunc of [:REAL,REAL:],REAL; :: thesis: for Pg2 being PartFunc of REAL,REAL st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) holds
( Pg2 is_integrable_on L-Meas & integral (Pg2,I) = Integral (L-Meas,Pg2) & integral (Pg2,I) = Integral (L-Meas,(ProjPMap2 (|.(R_EAL g).|,y))) & integral (Pg2,I) = (Integral1 (L-Meas,|.(R_EAL g).|)) . y )
let Pg2 be PartFunc of REAL,REAL; :: thesis: ( y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g & Pg2 = ProjPMap2 (|.(R_EAL g).|,y) implies ( Pg2 is_integrable_on L-Meas & integral (Pg2,I) = Integral (L-Meas,Pg2) & integral (Pg2,I) = Integral (L-Meas,(ProjPMap2 (|.(R_EAL g).|,y))) & integral (Pg2,I) = (Integral1 (L-Meas,|.(R_EAL g).|)) . y ) )
assume that
A1: y in J and
A2: dom f = [:I,J:] and
A3: f is_continuous_on [:I,J:] and
A4: f = g and
A5: Pg2 = ProjPMap2 (|.(R_EAL g).|,y) ; :: thesis: ( Pg2 is_integrable_on L-Meas & integral (Pg2,I) = Integral (L-Meas,Pg2) & integral (Pg2,I) = Integral (L-Meas,(ProjPMap2 (|.(R_EAL g).|,y))) & integral (Pg2,I) = (Integral1 (L-Meas,|.(R_EAL g).|)) . y )
A6: I is Element of L-Field by MEASUR10:5, MEASUR12:75;
A7: dom Pg2 = I by A1, A2, A4, A5, Th28;
( Pg2 | I is bounded & Pg2 is_integrable_on I ) by A1, A2, A3, A4, A5, Th47;
hence ( Pg2 is_integrable_on L-Meas & integral (Pg2,I) = Integral (L-Meas,Pg2) ) by A7, A6, MESFUN14:49; :: thesis: ( integral (Pg2,I) = Integral (L-Meas,(ProjPMap2 (|.(R_EAL g).|,y))) & integral (Pg2,I) = (Integral1 (L-Meas,|.(R_EAL g).|)) . y )
then integral (Pg2,I) = Integral (L-Meas,(ProjPMap2 (|.(R_EAL g).|,y))) by A5, MESFUNC5:def 7;
hence ( integral (Pg2,I) = Integral (L-Meas,(ProjPMap2 (|.(R_EAL g).|,y))) & integral (Pg2,I) = (Integral1 (L-Meas,|.(R_EAL g).|)) . y ) by MESFUN12:def 7; :: thesis: verum