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;

theorem Th5:
  {t1 where t1 is Element of T :
  for p st p in P holds not p is_a_proper_prefix_of t1} \
  {t1 where t1 is Element of T :
  for p st p in P holds not p is_a_prefix_of t1} = P
proof
 now
    let x be object;
    assume
A1: x in {t1 where t1 is Element of T :
    for p st p in P holds not p is_a_proper_prefix_of t1} \
    {t1 where t1 is Element of T :
    for p st p in P holds not p is_a_prefix_of t1};
then A2: x in {t1 where t1 is Element of T : for p st p in P holds not p
    is_a_proper_prefix_of t1};
A3: not x in {t1 where t1 is Element of T : for p st p in P holds not p
    is_a_prefix_of t1} by A1,XBOOLE_0:def 5;
    assume
A4: not x in P;
 ex t1 being Element of T st x = t1 &
    for p st p in P holds not p is_a_prefix_of t1
    proof
      consider t9 being Element of T such that
A5:   x = t9 and
A6:   for p st p in P holds not p is_a_proper_prefix_of t9 by A2;
   now
        let p;
        assume
A7:    p in P;
then A8:    not p is_a_proper_prefix_of t9 by A6;
        per cases by A8;
        suppose
      not p is_a_prefix_of t9;
          hence not p is_a_prefix_of t9;
        end;
        suppose
      p = t9;
          hence not p is_a_prefix_of t9 by A4,A5,A7;
        end;
      end;
      hence thesis by A5;
    end;
    hence contradiction by A3;
  end;
  hence {t1 where t1 is Element of T :
  for p st p in P holds not p is_a_proper_prefix_of t1} \
  {t1 where t1 is Element of T :
  for p st p in P holds not p is_a_prefix_of t1} c= P;
  let x be object;
  assume
A9: x in P;
A10: P c= {t1 where t1 is Element of T :
  for p st p in P holds not p is_a_proper_prefix_of t1} by Th4;
 not x in {t1 where t1 is Element of T :
  for p st p in P holds not p is_a_prefix_of t1}
  proof
    assume x in {t1 where t1 is Element of T :
    for p st p in P holds not p is_a_prefix_of t1};
then  ex t9 being Element of T st x = t9 & for p st p in P
    holds not p is_a_prefix_of t9;
    hence contradiction by A9;
  end;
  hence thesis by A9,A10,XBOOLE_0:def 5;
end;
