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

theorem Th2:
  for f be PartFunc of CNS,RNS st f is_uniformly_continuous_on X &
  X1 c= X holds f is_uniformly_continuous_on X1
proof
  let f be PartFunc of CNS,RNS;
  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 be Point of CNS 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 be Point of CNS;
  assume x1 in X1 & x2 in X1 & ||.x1-x2.||<s;
  hence thesis by A2,A4;
end;
