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 fT = 0 implies fT = elementary_tree 0
proof
  assume
A1: height fT = 0;
  thus fT c= elementary_tree 0
  proof
    let x be object;
    assume x in fT;
    then reconsider t = x as Element of fT;
 len t = 0 by A1,Def12;
then  x = {};
    hence thesis by Th21;
  end;
  let x be object;
  assume x in elementary_tree 0;
then  x = {} by Th28,TARSKI:def 1;
  hence thesis by Th21;
end;
