reserve n,m for Nat,
  x,X,X1 for set,
  s,g,r,p for Real,
  S,T for RealNormSpace,
  f,f1,f2 for PartFunc of S, T,
  s1,s2,q1 for sequence of S,
  x0,x1, x2 for Point of S,
  Y for Subset of S;

theorem
  f is_uniformly_continuous_on X implies ||.f .|| is_uniformly_continuous_on X
proof
  assume
A1: f is_uniformly_continuous_on X;
  then X c= dom f;
  hence
A2: X c= dom (||.f.||) by NORMSP_0:def 3;
  let r;
  assume 0<r;
  then consider s such that
A3: 0<s and
A4: for x1,x2 st x1 in X & x2 in X & ||.x1-x2.||<s holds ||.f/.x1-f/.x2
  .||<r by A1;
  take s;
  thus 0<s by A3;
  let x1,x2;
  assume that
A5: x1 in X and
A6: x2 in X and
A7: ||.x1-x2.||<s;
  |.(||.f.||)/.x1-(||.f.||)/.x2.| =|.(||.f.||).x1-(||.f.||)/.x2.| by A2,A5,
PARTFUN1:def 6
    .=|.(||.f.||).x1-(||.f.||).x2.| by A2,A6,PARTFUN1:def 6
    .= |.||.f/.x1.||-(||.f.||).x2.| by A2,A5,NORMSP_0:def 3
    .= |.||.f/.x1.|| - ||.f/.x2.||.| by A2,A6,NORMSP_0:def 3;
  then
A8: |.(||.f.||)/.x1-(||.f.||)/.x2.| <= ||.f/.x1-f/.x2.|| by NORMSP_1:9;
  ||.f/.x1-f/.x2.||<r by A4,A5,A6,A7;
  hence thesis by A8,XXREAL_0:2;
end;
