reserve n,m for Nat;
reserve x,X,X1 for set;
reserve s,g,r,p for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1,s2 for sequence of S;
reserve x0,x1,x2 for Point of S;
reserve Y for Subset of S;

theorem
  id Y is_Lipschitzian_on Y
proof
  reconsider r=1 as Real;
  thus Y c= dom(id Y) by RELAT_1:45;
  take r;
  thus r>0;
  let x1,x2;
  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;
