reserve x,y,z for object, X,Y for set,
  i,k,n for Nat,
  p,q,r,s for FinSequence,
  w for FinSequence of NAT,
  f for Function;

theorem Th52:
  tree({}) = elementary_tree 0
proof
  let p be FinSequence of NAT;
  thus p in tree({}) implies p in elementary_tree 0
  proof
    assume p in tree({});
    then
A1: p = {} or ex n,q st n < len {} & q in {} .(n+1) & p = <*n*>^q by Def15;
    assume not thesis;
    hence contradiction by A1,TARSKI:def 1,TREES_1:29;
  end;
  assume p in elementary_tree 0;
  then p = {} by TARSKI:def 1,TREES_1:29;
  hence thesis by Def15;
end;
