
theorem Th50:

:: 1.11. THEOREM, (2) => (1), p. 147
  for T being Lawson complete continuous TopLattice
  for S being infs-inheriting directed-sups-inheriting full non empty
  SubRelStr of T ex X being Subset of T st X = the carrier of S & X is closed
proof
  let T be Lawson complete continuous TopLattice;
  let S be infs-inheriting directed-sups-inheriting full non empty
  SubRelStr of T;
  reconsider X = the carrier of S as Subset of T by YELLOW_0:def 13;
  take X;
  thus X = the carrier of S;
  reconsider S as complete CLSubFrame of T by Th18;
  set SL = the Lawson correct TopAugmentation of S;
A1: the RelStr of SL = the RelStr of S by YELLOW_9:def 4;
  set f = incl(SL,T), f9 = incl(S,T);
A2: the carrier of S c= the carrier of T by YELLOW_0:def 13;
  then
A3: f = id the carrier of SL by A1,YELLOW_9:def 1;
A4: f9 = id the carrier of SL by A1,A2,YELLOW_9:def 1;
A5: [#]SL is compact by COMPTS_1:1;
A6: f9 is infs-preserving by Th8;
  the RelStr of T = the RelStr of T;
  then
A7: f is infs-preserving directed-sups-preserving by A1,A3,A4,A6,Th6,Th10;
  then f is SemilatticeHomomorphism of SL,T by Th5;
  then f is continuous by A7,Th46;
  then f.:[#]SL is compact by A5,WEIERSTR:8;
  then X is compact by A1,A3,FUNCT_1:92;
  hence thesis by COMPTS_1:7;
end;
