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 Th54:
  elementary_tree i = tree(i|->elementary_tree 0)
proof
  set p = i |-> elementary_tree 0;
  let q be FinSequence of NAT;
A1: len p = i by CARD_1:def 7;
  then elementary_tree i c= tree(p) by Th53;
  hence q in elementary_tree i implies q in tree(p);
  assume q in tree(p);
  then
A2: q = {} or
    ex n,r st n < len p & r in p.(n+1) & q = <*n*>^r by Def15;
  now
    given n, r such that
A3: n < len p and
A4: r in p.(n+1) and
A5: q = <*n*>^r;
    p = (Seg i) --> elementary_tree 0 by FINSEQ_2:def 2;
    then
A6: rng p c= {elementary_tree 0} by FUNCOP_1:13;
    p.(n+1) in rng p by A3,Lm3;
    then p.(n+1) = elementary_tree 0 by A6,TARSKI:def 1;
    then r = {} by A4,TARSKI:def 1,TREES_1:29;
    then q = <*n*> by A5,FINSEQ_1:34;
    hence thesis by A1,A3,TREES_1:28;
  end;
  hence thesis by A2,TREES_1:22;
end;
