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
  for T being Tree holds T is T-Substitution of 0
proof
  let T be Tree;
  let t be Element of T;
  assume
A1: not t in elementary_tree 0;
  set l = the Leaf of elementary_tree 0;
  take l;
A2: l = {} by TARSKI:def 1,TREES_1:29;
A3: t <> {} by A1,TARSKI:def 1,TREES_1:29;
  {} is_a_prefix_of t;
  hence thesis by A2,A3;
end;
