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 Th10:
  P<>{} implies for q st q in dom tree(T,P,T1) holds
  (for p st p in P holds not p is_a_prefix_of q & tree(T,P,T1).q = T.q)
  or ex p,r st p in P & r in dom T1 & q = p^r & tree(T,P,T1).q = T1.r
proof
  assume
A1: P<>{};
  let q;
  assume q in dom tree(T,P,T1);
then  q in tree(dom T,P,dom T1) by A1,Def2;
  hence thesis by Def2;
end;
