theorem Th3:
  {t1 where t1 is Element of T :
  for p st p in P holds not p is_a_prefix_of t1} 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 x in {t1 where t1 is Element of T :
  for p st p in P holds not p is_a_prefix_of t1};
  then consider t9 being Element of T such that
A1: x = t9 and
A2: for p st p in P holds not p is_a_prefix_of t9;
   for p st p in P holds not p is_a_proper_prefix_of t9 by A2;
  hence thesis by A1;
end;
