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 f is_continuous_on dom f & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) holds
Pg1 is continuous
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 f is_continuous_on dom f & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) holds
Pg1 is continuous
let g be PartFunc of [:[:REAL,REAL:],REAL:],REAL; :: thesis: for Pg1 being PartFunc of REAL,REAL st f is_continuous_on dom f & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) holds
Pg1 is continuous
let Pg1 be PartFunc of REAL,REAL; :: thesis: ( f is_continuous_on dom f & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) implies Pg1 is continuous )
assume that
A1: f is_continuous_on dom f and
A2: f = g and
A3: Pg1 = ProjPMap1 (|.(R_EAL g).|,[x,y]) ; :: thesis: Pg1 is continuous
Pg1 = |.(R_EAL (ProjPMap1 (g,[x,y]))).| by A3, MESFUN16:31;
then Pg1 = R_EAL (abs (ProjPMap1 (g,[x,y]))) by MESFUNC6:1;
then Pg1 = R_EAL (ProjPMap1 (|.g.|,[x,y])) by MESFUN16:32;
then Pg1 = ProjPMap1 (|.g.|,[x,y]) by MESFUNC5:def 7;
hence Pg1 is continuous by A1, A2, Th13; :: thesis: verum