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 Th9:
  f is_Lipschitzian_on X implies f is_uniformly_continuous_on X
proof
  assume
A1: f is_Lipschitzian_on X;
  hence X c= dom f by NFCONT_1:def 9;
  consider r be Real such that
A2: 0<r and
A3: for x1,x2 st x1 in X & x2 in X holds ||.f/.x1-f/.x2.||<=r*||.x1-x2
  .|| by A1,NFCONT_1:def 9;
  let p be Real such that
A4: 0<p;
  take s=p/r;
  thus 0<s by A2,A4,XREAL_1:139;
  let x1,x2;
  assume x1 in X & x2 in X & ||.x1 -x2.|| < s;
  then r*||.x1 -x2.|| < s*r & ||.f/.x1-f/.x2.||<=r*||.x1-x2.|| by A2,A3,
XREAL_1:68;
  then ||.f/.x1-f/.x2.||<p/r*r by XXREAL_0:2;
  hence thesis by A2,XCMPLX_1:87;
end;
