reserve
  X,x,y,z for set,
  k,n,m for Nat ,
  f for Function,
  p,q,r for FinSequence of NAT;
reserve p1,p2,p3 for FinSequence;
reserve T,T1 for Tree;
reserve fT,fT1 for finite Tree;

theorem
  p in elementary_tree n implies p = {} or ex k st k < n & p = <*k*>
proof
  assume p in elementary_tree n;
then A1: p in D or p in { <*k*> : k < n } by XBOOLE_0:def 3;
 p in { <*k*> : k < n } implies ex k st p = <*k*> & k < n;
  hence thesis by A1,TARSKI:def 1;
end;
