reserve x, y, z for object,
  i, j, n for Nat,
  p, q, r for FinSequence,
  v for FinSequence of NAT;

theorem Th8:
 for j being Element of NAT st  j < i
  holds (elementary_tree i)|<*j*> = elementary_tree 0
proof let j be Element of NAT;
  set p = i |-> elementary_tree 0, T = tree(p);
  assume
A1: j < i;
then A2: 1+j >= 1 & j+1 <= i by NAT_1:11,13;
 len p = i by CARD_1:def 7;
  then A3: elementary_tree i = T & T|<*j*> = p.(j+1) by A1,TREES_3:49,54;
 j+1 in Seg i by A2;
  hence thesis by A3,FUNCOP_1:7;
end;
