
theorem Th38:
 for T being Lawson complete TopLattice
 for x being Element of T holds
   uparrow x is closed & downarrow x is closed & {x} is closed
  proof let T be Lawson complete TopLattice;
   set S = the Scott TopAugmentation of T;
   set R = the lower correct TopAugmentation of T;
A:  the RelStr of S = the RelStr of T & the RelStr of R = the RelStr of T
     by YELLOW_9:def 4;
    T is TopAugmentation of T by YELLOW_9:44; then
C:  T is Refinement of S,R by Th29; then
D:  T is Refinement of R,S by YELLOW_9:55;
   let x be Element of T;
   reconsider y = x as Element of S by A;
   reconsider z = x as Element of R by A;
   downarrow y = downarrow x & downarrow y is closed &
   uparrow z = uparrow x & uparrow z is closed
   by A,WAYBEL_0:13,Th4,WAYBEL11:11;
   hence uparrow x is closed & downarrow x is closed by A,C,D,Th21;
   then (uparrow x) /\ downarrow x is closed;
   hence thesis by Th28;
  end;
