reserve n,m for Nat;
reserve x,X,X1 for set;
reserve s,g,r,p for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1,s2 for sequence of S;
reserve x0,x1,x2 for Point of S;
reserve Y for Subset of S;

theorem Th53:
  for f be PartFunc of the carrier of S,REAL st (for x0 st x0 in
  dom f holds f/.x0 = ||. x0.|| ) holds f is_continuous_on (dom f)
proof
  let f be PartFunc of the carrier of S,REAL;
  assume
A1: for x0 st x0 in dom f holds f/.x0 = ||. x0 .||;
  now
    let x1,x2;
    assume x1 in dom f & x2 in dom f;
    then f/.x1 = ||. x1 .|| & f/.x2 = ||. x2.|| by A1;
    hence |.f/.x1-f/.x2.| <= 1*||. x1-x2.|| by NORMSP_1:9;
  end;
  then f is_Lipschitzian_on dom f;
  hence thesis by Th46;
end;
