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 Th8:
  for f be PartFunc of the carrier of S, REAL st f
  is_uniformly_continuous_on X holds f is_continuous_on X
proof
  let f be PartFunc of the carrier of S, REAL;
  assume
A1: f is_uniformly_continuous_on X;
A2: now
    let x0;
    let r be Real;
    assume that
A3: x0 in X and
A4: 0<r;
    consider s be Real such that
A5: 0<s and
A6: for x1,x2 st x1 in X & x2 in X & ||.x1 -x2.|| < s holds |.f/.x1
    - f/.x2.|< r by A1,A4;
    take s;
    thus 0<s by A5;
    let x1;
    assume x1 in X & ||.x1-x0.|| < s;
    hence |.f/.x1 - f/.x0.| < r by A3,A6;
  end;
  X c= dom f by A1;
  hence thesis by A2,NFCONT_1:20;
end;
