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
  for Y be Subset of CNS holds id Y is_Lipschitzian_on Y
proof
  reconsider r=1 as Real;
  let Y be Subset of CNS;
  thus Y c= dom(id Y) by RELAT_1:45;
  take r;
  thus r>0;
  let x1,x2 be Point of CNS;
  assume that
A1: x1 in Y and
A2: x2 in Y;
  ||. (id Y)/.x1-(id Y)/.x2 .|| = ||. x1-(id Y)/.x2 .|| by A1,PARTFUN2:6
    .= r*||. x1-x2 .|| by A2,PARTFUN2:6;
  hence thesis;
end;
