reserve x for set;

theorem Th21:
  for L being complete LATTICE holds omega L c= xi L
proof
  let L be complete LATTICE;
  set S = the lower correct TopAugmentation of L;
  set X = the lim-inf TopAugmentation of L;
  reconsider B = the set of all (uparrow x)` where x is Element of S as
  prebasis of S by WAYBEL19:def 1;
A1: the RelStr of S = the RelStr of L & the RelStr of X = the RelStr of L by
YELLOW_9:def 4;
A2: B c= the topology of X
  proof
    let b be object;
    assume b in B;
    then consider x being Element of S such that
A3: b = (uparrow x)`;
    reconsider y = x as Element of X by A1;
    set A = uparrow y;
    X is SubRelStr of X by YELLOW_6:6;
    then S is SubRelStr of X by A1,WAYBEL21:12;
    then
A4: uparrow x c= uparrow y by WAYBEL23:14;
A5: inf A = y by WAYBEL_0:39;
    now
      let F be ultra Filter of BoolePoset [#]X;
      assume A in F;
      then inf A in {inf C where C is Subset of X: C in F};
      then inf A <= "\/"({inf C where C is Subset of X: C in F}, X) by
YELLOW_2:22;
      hence lim_inf F in A by A5,WAYBEL_0:18;
    end;
    then
A6: A is closed by Th18;
    S is SubRelStr of S by YELLOW_6:6;
    then X is SubRelStr of S by A1,WAYBEL21:12;
    then uparrow y c= uparrow x by WAYBEL23:14;
    then uparrow y = uparrow x by A4;
    hence thesis by A1,A3,A6,PRE_TOPC:def 2;
  end;
  the carrier of S in the topology of X by A1,PRE_TOPC:def 1;
  then the topology of S c= the topology of X by A2,Th20;
  then omega L c= the topology of X by WAYBEL19:def 2;
  hence thesis by Th10;
end;
