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 f is_continuous_on dom f & f = g & Pg2 = ProjPMap2 ((R_EAL g),y) holds
Pg2 is continuous
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 f is_continuous_on dom f & f = g & Pg2 = ProjPMap2 ((R_EAL g),y) holds
Pg2 is continuous
let g be PartFunc of [:REAL,REAL:],REAL; :: thesis: for Pg2 being PartFunc of REAL,REAL st f is_continuous_on dom f & f = g & Pg2 = ProjPMap2 ((R_EAL g),y) holds
Pg2 is continuous
let Pg2 be PartFunc of REAL,REAL; :: thesis: ( f is_continuous_on dom f & f = g & Pg2 = ProjPMap2 ((R_EAL g),y) implies Pg2 is continuous )
assume that
A1: f is_continuous_on dom f and
A2: f = g and
A3: Pg2 = ProjPMap2 ((R_EAL g),y) ; :: thesis: Pg2 is continuous
Pg2 = R_EAL (ProjPMap2 (g,y)) by A3, Th31;
then Pg2 = ProjPMap2 (g,y) by MESFUNC5:def 7;
hence Pg2 is continuous by A1, A2, Th33; :: thesis: verum