let n be non zero Nat; :: thesis: for f being PartFunc of [:(CarProduct ((Seg n) --> RNS_Real)),RNS_Real:],RNS_Real
for g being PartFunc of [:(CarProduct ((Seg n) --> REAL)),REAL:],REAL
for Pg1 being PartFunc of REAL,REAL
for x being Element of CarProduct ((Seg n) --> REAL) st f is_continuous_on dom f & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) holds
Pg1 is continuous
let f be PartFunc of [:(CarProduct ((Seg n) --> RNS_Real)),RNS_Real:],RNS_Real; :: thesis: for g being PartFunc of [:(CarProduct ((Seg n) --> REAL)),REAL:],REAL
for Pg1 being PartFunc of REAL,REAL
for x being Element of CarProduct ((Seg n) --> REAL) st f is_continuous_on dom f & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) holds
Pg1 is continuous
let g be PartFunc of [:(CarProduct ((Seg n) --> REAL)),REAL:],REAL; :: thesis: for Pg1 being PartFunc of REAL,REAL
for x being Element of CarProduct ((Seg n) --> REAL) st f is_continuous_on dom f & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) holds
Pg1 is continuous
let Pg1 be PartFunc of REAL,REAL; :: thesis: for x being Element of CarProduct ((Seg n) --> REAL) st f is_continuous_on dom f & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) holds
Pg1 is continuous
let x be Element of CarProduct ((Seg n) --> REAL); :: thesis: ( f is_continuous_on dom f & f = g & Pg1 = ProjPMap1 (|.(R_EAL g).|,x) 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) ; :: thesis: Pg1 is continuous
Pg1 = |.(R_EAL (ProjPMap1 (g,x))).| by A3, MESFUN16:31;
then Pg1 = R_EAL (abs (ProjPMap1 (g,x))) by MESFUNC6:1;
then Pg1 = R_EAL (ProjPMap1 (|.g.|,x)) by MESFUN16:32;
then Pg1 = ProjPMap1 (|.g.|,x) by MESFUNC5:def 7;
hence Pg1 is continuous by A1, A2, Th18; :: thesis: verum