reserve x for set;

theorem Th24: :: III. 3.5. LEMMA, p. 168(?)
  for L being complete LATTICE holds lambda L c= xi L
proof
  let L be complete LATTICE;
  set T = the Lawson correct TopAugmentation of L;
  set S = the Scott TopAugmentation of L;
  set LL = the lower correct TopAugmentation of L;
  set LI = the lim-inf TopAugmentation of L;
A1: the RelStr of LI = the RelStr of L by YELLOW_9:def 4;
A2: xi L = the topology of LI by Th10;
  omega L = the topology of LL by WAYBEL19:def 2;
  then the RelStr of LL = the RelStr of L & the topology of LL c= xi L by Th21,
YELLOW_9:def 4;
  then
A3: LI is TopExtension of LL by A2,A1,YELLOW_9:def 5;
  sigma L = the topology of S by YELLOW_9:51;
  then the RelStr of S = the RelStr of L & the topology of S c= xi L by Th19,
YELLOW_9:def 4;
  then T is Refinement of S,LL & LI is TopExtension of S by A2,A1,WAYBEL19:29
,YELLOW_9:def 5;
  then lambda L = the topology of T & LI is TopExtension of T by A3,Th22,
WAYBEL19:def 4;
  hence thesis by A2,YELLOW_9:def 5;
end;
