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 Th4:
  P c= {t1 where t1 is Element of T :
  for p st p in P holds not p is_a_proper_prefix_of t1}
proof
  let x be object;
  assume
A1: x in P;
 ex t1 being Element of T st x = t1 &
  for p st p in P holds not p is_a_proper_prefix_of t1
  proof
 P c= T by TREES_1:def 11;
    then consider t9 being Element of T such that
A2: t9 = x by A1;
 now
      let p such that
A3:   p in P;
      per cases;
      suppose t9 = p;
        hence not p is_a_proper_prefix_of t9;
      end;
      suppose t9 <> p;
        then not t9, p are_c=-comparable by A1,A2,A3,TREES_1:def 10;
        hence not p is_a_proper_prefix_of t9;
      end;
    end;
    hence thesis by A2;
  end;
  hence thesis;
end;
