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

theorem
  for p being DTree-yielding FinSequence, n being Nat,
  q being FinSequence st <*n*>^q in dom (x-tree(p)) holds
  (x-tree(p)).(<*n*>^q) = p..(n+1,q)
proof
  let p be DTree-yielding FinSequence, n be Nat, q be FinSequence;
  assume
A1: <*n*>^q in dom (x-tree(p));
then  <*n*>^q is Node of (x-tree(p));
  then reconsider q9 = q as FinSequence of NAT by FINSEQ_1:36;
  reconsider n as Element of NAT by ORDINAL1:def 12;
A2: <*n*> in dom (x-tree p) by A1,TREES_1:21;
A3: <*n*>^q in tree(doms p) by A1,Th10;
A4: len doms p = len p by TREES_3:38;
A5: q9 in (dom (x-tree p))|<*n*> by A1,A2,TREES_1:def 6;
A6: n < len p by A3,A4,TREES_3:48;
  A7: dom
 ((x-tree p)|<*n*>) = (dom (x-tree p))|<*n*> & ((x-tree(p))|<*n*>).q9 = (
  x-tree(p)).(<*n*>^q) by A5,TREES_2:def 10;
 n+1 in dom p & p.(n+1) = (x-tree(p))|<*n*> by A6,Def4,Lm2;
  hence thesis by A5,A7,FUNCT_5:38;
end;
