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

theorem Th12:
  for p being DTree-yielding FinSequence for i being Nat,
  T being DecoratedTree, q being Node of T st i < len p &
  T = p.(i+1) holds (x-tree p).(<*i*>^q) = T.q
proof
  let p be DTree-yielding FinSequence, n be Nat,
  T be DecoratedTree;
  let q be Node of T;
  assume
A1: n < len p & T = p.(n+1);
   reconsider n as Element of NAT by ORDINAL1:def 12;
A2: <*n*>^q in dom (x-tree(p)) by Th11,A1;
  then <*n*> in dom (x-tree(p)) by TREES_1:21;
then  q in (dom (x-tree p))|<*n*> by A2,TREES_1:def 6;
then  ((x-tree(p))|<*n*>).q = (x-tree(p)).(<*n*>^q) by TREES_2:def 10;
  hence thesis by A1,Def4;
end;
