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

theorem Th9:
 for i being Element of NAT st i < len p
 holds (x-flat_tree p)|<*i*> = root-tree (p.(i+1))
proof let i be Element of NAT;
  reconsider t = {} as Element of (dom (x-flat_tree p))|<*i*> by TREES_1:22;
  assume
A1: i < len p;
then A2: (x-flat_tree p).<*i*> = p.(i+1) by Def3;
A3: dom (x-flat_tree p) = elementary_tree len p by Def3;
 (elementary_tree len p)|<*i*> = elementary_tree 0 by A1,Th8;
then
A4: dom ((x-flat_tree p)|<*i*>) = elementary_tree 0 by A3,TREES_2:def 10;
   <*
i*>^t = <*i*> & ((x-flat_tree p)|<*i*>).t = (x-flat_tree p).(<*i*>^t) by
FINSEQ_1:34,TREES_2:def 10;
  hence thesis by A2,A4,Th5;
end;
