reserve x for set;

theorem Th26:
  for L being complete lim-inf TopLattice for F being ultra Filter
  of BoolePoset [#]L holds lim_inf F is_a_convergence_point_of F, L
proof
  let L be complete lim-inf TopLattice;
  let F be ultra Filter of BoolePoset [#]L;
  set x = lim_inf F;
  let A be Subset of L;
  assume that
A1: A is open and
A2: x in A and
A3: not A in F;
  F is prime by WAYBEL_7:22;
  then
A4: ([#]L)\A in F by A3,WAYBEL_7:21;
  then A` <> {} by YELLOW19:1;
  then x in A` by A1,A4,Th18;
  hence contradiction by A2,XBOOLE_0:def 5;
end;
