
theorem Th30:
  for T being Lawson complete TopLattice holds (sigma T) \/ the set of all (
  uparrow x)` where x is Element of T is prebasis of T
proof
  let T be Lawson complete TopLattice;
  set R = the lower correct TopAugmentation of T;
  set S = the Scott TopAugmentation of T;
A1: the RelStr of S = the RelStr of T by YELLOW_9:def 4;
  T is TopAugmentation of T by YELLOW_9:44;
  then
A2: T is Refinement of S,R by Th29;
  set K = the set of all (uparrow x)` where x is Element of R;
  set O = the set of all (uparrow x)` where x is Element of T;
  the topology of S = sigma T by YELLOW_9:51;
  then sigma T is Basis of S by CANTOR_1:2;
  then
A3: sigma T is prebasis of S by YELLOW_9:27;
A4: the RelStr of R = the RelStr of T by YELLOW_9:def 4;
  O = K
  proof
    hereby
      let a be object;
      assume a in O;
      then consider x being Element of T such that
A5:   a = (uparrow x)`;
      reconsider y = x as Element of R by A4;
      uparrow x = uparrow y by A4,WAYBEL_0:13;
      hence a in K by A4,A5;
    end;
    let a be object;
    assume a in K;
    then consider x being Element of R such that
A6: a = (uparrow x)`;
    reconsider y = x as Element of T by A4;
    uparrow x = uparrow y by A4,WAYBEL_0:13;
    hence thesis by A4,A6;
  end;
  then reconsider O as prebasis of R by Def1;
  O = O;
  hence thesis by A3,A2,A1,A4,Th23;
end;
