reserve n,m for Element of NAT;
reserve r,s for Real;
reserve z for Complex;
reserve CNS,CNS1,CNS2 for ComplexNormSpace;
reserve RNS for RealNormSpace;
reserve X,X1 for set;

theorem Th99:
  for f be PartFunc of RNS,CNS st f is_Lipschitzian_on X & X1 c=
  X holds f is_Lipschitzian_on X1
proof
  let f be PartFunc of RNS,CNS;
  assume that
A1: f is_Lipschitzian_on X and
A2: X1 c= X;
  X c= dom f by A1;
  hence X1 c= dom f by A2;
  consider s be Real such that
A3: 0<s and
A4: for x1,x2 be Point of RNS st x1 in X & x2 in X holds ||. f/.x1-f/.x2
  .||<=s*||. x1-x2.|| by A1;
  take s;
  thus 0<s by A3;
  let x1,x2 be Point of RNS;
  assume x1 in X1 & x2 in X1;
  hence thesis by A2,A4;
end;
