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 Th12:
  P<>{} implies for q st q in dom tree(T,P,T1) &
  q in {t1 where t1 is Element of dom T :
  for p st p in P holds not p is_a_prefix_of t1} holds tree(T,P,T1).q = T.q
proof
  assume
A1: P<>{};
  let q;
  assume that
A2: q in dom tree(T,P,T1) and
A3: q in {t1 where t1 is Element of dom T : for p st p in P holds not p
  is_a_prefix_of t1};
A4: ex t9 being Element of dom T st t9 = q & for p st p in P
  holds not p is_a_prefix_of t9 by A3;
  per cases by A2,Th10;
  suppose
A5: for p st p in P holds not p is_a_prefix_of q & tree(T,P,T1).q = T.q;
    consider x being object such that
A6: x in P by A1,XBOOLE_0:def 1;
 P c= dom T by TREES_1:def 11;
    then reconsider x as Element of dom T by A6;
 ex p9 st p9 = x;
    hence thesis by A5,A6;
  end;
  suppose
     ex p,r st p in P & r in dom T1 & q = p^r & tree(T,P,T1).q = T1.r;
    then consider p,r such that
A7: p in P and r in dom T1 and
A8: q = p^r and tree(T,P,T1).q = T1.r;
 not p is_a_prefix_of q by A4,A7;
    hence thesis by A8,TREES_1:1;
  end;
end;
