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

theorem Th19:
  for f be PartFunc of CNS1,CNS2 st f is_uniformly_continuous_on X
  holds f is_continuous_on X
proof
  let f be PartFunc of CNS1,CNS2;
  assume
A1: f is_uniformly_continuous_on X;
A2: now
    let x0 be Point of CNS1;
    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 be Point of CNS1 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 be Point of CNS1;
    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,NCFCONT1:44;
end;
