reserve x for set;

theorem :: 3.6. PROPOSITION (A), p. 169(?)
  for L being complete lim-inf TopLattice holds L is compact T_1
proof
  let L be complete lim-inf TopLattice;
  set T = the Lawson correct TopAugmentation of L;
  now
    let F be ultra Filter of BoolePoset [#]L;
    reconsider x = lim_inf F as Point of L;
    take x;
    thus x is_a_convergence_point_of F, L by Th26;
  end;
  hence L is compact by YELLOW19:31;
  now
    let x be Point of L;
    reconsider y = x as Element of L;
    the RelStr of L = the RelStr of T by YELLOW_9:def 4;
    then reconsider z = y as Element of T;
    L is TopAugmentation of L by YELLOW_9:44;
    then
A1: L is TopExtension of T by Th25;
    {z} is closed;
    then {y} is closed by A1,Th23;
    hence Cl {x} = {x} by PRE_TOPC:22;
  end;
  hence thesis by FRECHET2:43;
end;
