reserve X,X1 for set,
  r,s for Real,
  z for Complex,
  RNS for RealNormSpace,
  CNS, CNS1,CNS2 for ComplexNormSpace;

theorem
  for f be PartFunc of RNS,CNS st f is_uniformly_continuous_on X holds
  ||.f.|| is_uniformly_continuous_on X
proof
  let f be PartFunc of RNS,CNS;
  assume
A1: f is_uniformly_continuous_on X;
  then X c= dom f;
  then
A2: X c= dom (||.f.||) by NORMSP_0:def 3;
  for r be Real st 0 < r ex s be Real st 0 < s &
    for x1,x2 be Point of RNS
st x1 in X & x2 in X & ||.x1-x2.|| < s holds |.(||.f.||)/.x1 - (||.f.||)/.x2
  .| < r
  proof
    let r be Real;
    assume 0 < r;
    then consider s be Real such that
A3: 0 < s and
A4: for x1,x2 be Point of RNS st x1 in X & x2 in X & ||.x1-x2.|| < s
    holds ||. f/.x1 - f/.x2 .|| < r by A1;
     reconsider s as Real;
    take s;
    thus 0<s by A3;
    let x1,x2 be Point of RNS;
    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 CLVECT_1:110;
    ||.f/.x1-f/.x2.||<r by A4,A5,A6,A7;
    hence thesis by A8,XXREAL_0:2;
  end;
  hence thesis by A2,NFCONT_2:def 2;
end;
