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 p2 being PartFunc of REAL,REAL st f is_continuous_on dom f & f = g & p2 = ProjPMap2 (|.(R_EAL g).|,y) holds
p2 is continuous
let f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real; :: thesis: for g being PartFunc of [:REAL,REAL:],REAL
for p2 being PartFunc of REAL,REAL st f is_continuous_on dom f & f = g & p2 = ProjPMap2 (|.(R_EAL g).|,y) holds
p2 is continuous
let g be PartFunc of [:REAL,REAL:],REAL; :: thesis: for p2 being PartFunc of REAL,REAL st f is_continuous_on dom f & f = g & p2 = ProjPMap2 (|.(R_EAL g).|,y) holds
p2 is continuous
let p2 be PartFunc of REAL,REAL; :: thesis: ( f is_continuous_on dom f & f = g & p2 = ProjPMap2 (|.(R_EAL g).|,y) implies p2 is continuous )
assume that
A1: f is_continuous_on dom f and
A2: f = g and
A3: p2 = ProjPMap2 (|.(R_EAL g).|,y) ; :: thesis: p2 is continuous
p2 = |.(R_EAL (ProjPMap2 (g,y))).| by A3, Th31;
then p2 = R_EAL (abs (ProjPMap2 (g,y))) by MESFUNC6:1;
then p2 = R_EAL (ProjPMap2 (|.g.|,y)) by Th32;
then p2 = ProjPMap2 (|.g.|,y) by MESFUNC5:def 7;
hence p2 is continuous by A1, A2, Th34; :: thesis: verum