
theorem Th21:
for S be RealNormSpace, x0 be Point of S, f,g be PartFunc of S,RNS_Real st
 f is_continuous_in x0 & g = ||.f.|| holds g is_continuous_in x0
proof
    let S be RealNormSpace, x0 be Point of S, f,g be PartFunc of S,RNS_Real;
    assume that
A1:  f is_continuous_in x0 and
A2:  g = ||.f.||;

A3: ||.f.|| is_continuous_in x0 by A1,NFCONT_1:17;

    for s1 be sequence of S st rng s1 c= dom g & s1 is convergent & lim s1 = x0
     holds g/*s1 is convergent & g/.x0 = lim(g/*s1)
    proof
     let s1 be sequence of S;
     assume that
A4:  rng s1 c= dom g and
A5:  s1 is convergent and
A6:  lim s1 = x0;

     ||.f.||/*s1 is convergent & ||.f.||/.x0 = lim(||.f.||/*s1)
       by A3,A2,A4,A5,A6;
     hence g/*s1 is convergent & g/.x0 = lim(g/*s1) by A2,DUALSP03:5,6;
    end;
    hence g is_continuous_in x0 by A2,A1,NORMSP_0:def 2;
end;
