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

theorem Th11:
  for p being DTree-yielding FinSequence holds
  y in dom (x-tree(p)) iff y = {} or
  ex i being Nat, T being DecoratedTree, q being Node of T st
  i < len p & T = p.(i+1) & y = <*i*>^q
proof
  let p be DTree-yielding FinSequence;
A1: dom (x-tree p) = tree(doms p) by Th10;
A2: now
    given i,q such that
A3: i < len doms p and
A4: q in (doms p).(i+1) and
A5: y = <*i*>^q;
 len doms p = len p by TREES_3:38;
then A6: i+1 in dom p by A3,Lm2;
    then reconsider T = p.(i+1) as DecoratedTree by TREES_3:24;
    take i, T;
    reconsider q as Node of T by A4,A6,FUNCT_6:22;
    take q;
    thus i < len p & T = p.(i+1) & y = <*i*>^q by A3,A5,TREES_3:38;
  end;
 now
    given i being Nat, T being DecoratedTree, q being Node of T
    such that
A7: i < len p and
A8: T = p.(i+1) and
A9: y = <*i*>^q;
    reconsider q as FinSequence;
    take i, q;
 i+1 in dom p by A7,Lm2;
then  (doms p).(i+1) = dom T by A8,FUNCT_6:22;
    hence i < len doms p & q in (doms p).(i+1) & y = <*i*>^q by A7,A9,
TREES_3:38;
  end;
  hence thesis by A1,A2,TREES_3:def 15;
end;
