reserve T, T1 for Tree,
  P for AntiChain_of_Prefixes of T,
  p1 for FinSequence,
  p, q, r, s, p9 for FinSequence of NAT,
  x, Z for set,
  t for Element of T,
  k, n for Nat;
reserve T,T1 for DecoratedTree,
  P for AntiChain_of_Prefixes of dom T,
  t for Element of dom T,
  p1, p2, r1, r2 for FinSequence of NAT;

theorem Th11:
  p in dom T implies for q st q in dom (T with-replacement (p,T1)) holds
  not p is_a_prefix_of q & T with-replacement (p,T1).q = T.q
  or ex r st r in dom T1 & q = p^r & T with-replacement (p,T1).q = T1.r
proof
  assume
A1: p in dom T;
  let q;
  assume q in dom (T with-replacement (p,T1));
then  q in dom T with-replacement (p,dom T1) by A1,TREES_2:def 11;
  hence thesis by A1,TREES_2:def 11;
end;
