reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th43:
for f be PartFunc of REAL m,REAL,
    g be PartFunc of REAL-NS m,REAL
 st f = g & Z c= dom f holds
   f is_continuous_on Z
 iff
   for s be sequence of REAL-NS m st rng s c= Z & s is convergent & lim s in Z
      holds g/*s is convergent & g/.(lim s) = lim (g/*s)
proof
   let f be PartFunc of REAL m,REAL,
       g be PartFunc of REAL-NS m,REAL;
   assume A1: f = g;
   assume A2: Z c= dom f;
hereby assume f is_continuous_on Z; then
    g is_continuous_on Z by A2,Th42,A1;
    hence for s1 be sequence of REAL-NS m
    st rng s1 c= Z & s1 is convergent & lim s1 in Z
      holds g/*s1 is convergent & g/.(lim s1) = lim (g/*s1) by Th2;
   end;
   assume for s be sequence of REAL-NS m st rng s c= Z
     & s is convergent & lim s in Z
       holds g/*s is convergent & g/.(lim s) = lim (g/*s); then
    g is_continuous_on Z by A1,A2,Th2;
    hence f is_continuous_on Z by Th42,A1;
end;
