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;
reserve t for Element of T;
reserve w for FinSequence;
reserve t1,t2 for Element of T;

theorem
  height elementary_tree(n+1) = 1
proof
  set T = elementary_tree(n+1);
 now
    thus ex p st p in T & len p = 1
    proof
      take p = <*0*>;
      thus p in T by Th27;
      thus thesis by FINSEQ_1:40;
    end;
    let p such that
A1: p in T;
A2: p in D implies p = {} by TARSKI:def 1;
 now
      assume p in { <*k*> : k < n+1 };
then    ex k st p = <*k*> & k < n+1;
      hence len p = 1 by FINSEQ_1:40;
    end;
    hence len p <= 1 by A1,A2,XBOOLE_0:def 3;
  end;
  hence thesis by Def12;
end;
