
theorem Th26:
  for L being non trivial Boolean LATTICE, F being proper Filter
  of L ex G being Filter of L st F c= G & G is ultra
proof
  let L be non trivial Boolean LATTICE;
  let F be proper Filter of L;
  downarrow Bottom L = {Bottom L} by WAYBEL_4:23;
  then reconsider I = {Bottom L} as Ideal of L;
  now
    let a be object;
    assume a in I;
    then a = Bottom L by TARSKI:def 1;
    hence not a in F by Th4;
  end;
  then I misses F by XBOOLE_0:3;
  then consider G being Filter of L such that
A1: G is prime & F c= G and
A2: I misses G by Th25;
  take G;
  now
    assume Bottom L in G;
    then not Bottom L in I by A2,XBOOLE_0:3;
    hence contradiction by TARSKI:def 1;
  end;
  then G is proper;
  hence thesis by A1,Th22;
end;
