
theorem Th4:
  for L being lower-bounded non empty Poset, F being Filter of L
  holds F is proper iff not Bottom L in F
proof
  let L be lower-bounded non empty Poset, F be Filter of L;
  hereby
    assume F is proper;
    then F <> the carrier of L;
    then consider a being object such that
A1: not (a in F iff a in the carrier of L) by TARSKI:2;
    reconsider a as Element of L by A1;
    a >= Bottom L by YELLOW_0:44;
    hence not Bottom L in F by A1,WAYBEL_0:def 20;
  end;
  assume not Bottom L in F;
  then F <> the carrier of L;
  hence thesis;
end;
