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

theorem
  for T being DecoratedTree holds
  x-tree(<*T*>) = ((elementary_tree 1) --> x) with-replacement (<*0*>, T)
proof
  let T be DecoratedTree;
  set D = ((elementary_tree 1) --> x) with-replacement (<*0*>, T);
  set W = elementary_tree 1 with-replacement(<*0*>,dom T);
A1: dom (x-tree <*T*>) = tree(doms <*T*>) by Th10
    .= tree(<*dom T*>) by FINSEQ_3:132
    .= ^dom T by TREES_3:def 16
    .= W by TREES_3:58;
A2: dom ((elementary_tree 1) --> x) = elementary_tree 1;
  reconsider t1 = {}, t2 = <*0*> as Element of elementary_tree 1
  by TARSKI:def 2,TREES_1:51;
 t2 = t2;
then A3: dom D = W by A2,TREES_2:def 11;
A4: {} in dom T by TREES_1:22;
 now
    let y be object;
    assume y in W;
    then reconsider q = y as Element of W;
     q
 in elementary_tree 1 or ex v st v in dom T & q = t2^v by TREES_1:def 9;
then A5: q = {} or q = t2 & t2 = t2^t1 or
    ex v st v in dom T & q = <*0*>^v by FINSEQ_1:34,TARSKI:def 2,TREES_1:51;
 not t2 is_a_prefix_of t1;
then A6: D.{} = ((elementary_tree 1)-->x).t1 by A2,TREES_3:45
      .= x
      .= (x-tree <*T*>).{} by Def4;
 now
      given r being FinSequence of NAT such that
A7:  r in dom T and
A8:  q = <*0*>^r;
      reconsider r as Node of T by A7;
  q = t2^r by A8;
then A9:  D.q = T.r by A2,TREES_3:46;
  len <*T*> = 1 & <*T*>.(0+1) = T by FINSEQ_1:40;
then A10:  (x-tree <*T*>)|t2 = T by Def4;
  W|t2 = dom T by TREES_1:33;
      hence D.q = (x-tree <*T*>).q by A1,A8,A9,A10,TREES_2:def 10;
    end;
    hence D.y = (x-tree <*T*>).y by A4,A5,A6;
  end;
  hence thesis by A1,A3;
end;
