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 Th1:
  f is_uniformly_continuous_on X & X1 c= X implies f
  is_uniformly_continuous_on X1
proof
  assume that
A1: f is_uniformly_continuous_on X and
A2: X1 c= X;
  X c= dom f by A1;
  hence X1 c= dom f by A2,XBOOLE_1:1;
  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 x1 in X1 & x2 in X1 & ||.x1-x2.||<s;
  hence thesis by A2,A4;
end;
