
theorem
  for T being Lawson complete TopLattice holds (sigma T) \/ {W\uparrow
  x where W is Subset of T, x is Element of T: W in sigma T} is prebasis of T
proof
  let T be Lawson complete TopLattice;
  set R = the lower correct TopAugmentation of T;
  reconsider K = the set of all (uparrow x)` where x is Element of R as
  prebasis of R by Def1;
  set O = {W\uparrow x where W is Subset of T,x is Element of T: W in sigma T};
  O c= bool the carrier of T
  proof
    let a be object;
    assume a in O;
    then
    ex W being Subset of T, x being Element of T st a = W\uparrow x & W in
    sigma T;
    hence thesis;
  end;
  then reconsider O as Subset-Family of T;
  reconsider O as Subset-Family of T;
  set S = the Scott TopAugmentation of T;
A1: the RelStr of R = the RelStr of T by YELLOW_9:def 4;
  (sigma T) \/ omega T is prebasis of T by Def3;
  then
A2: (sigma T) \/ omega T c= the topology of T by TOPS_2:64;
  omega T c= (sigma T) \/ omega T by XBOOLE_1:7;
  then
A3: omega T c= the topology of T by A2;
  sigma T c= (sigma T) \/ omega T by XBOOLE_1:7;
  then
A4: sigma T c= the topology of T by A2;
A5: omega T = the topology of R by Def2;
  O c= the topology of T
  proof
    let a be object;
    assume a in O;
    then consider W being Subset of T, x being Element of T such that
A6: a = W \ uparrow x and
A7: W in sigma T;
A8: a = W /\ (uparrow x)` by A6,SUBSET_1:13;
    reconsider y = x as Element of R by A1;
    uparrow x = uparrow y by A1,WAYBEL_0:13;
    then
A9: (uparrow x)` in K by A1;
    K c= omega T by A5,TOPS_2:64;
    then (uparrow x)` in omega T by A9;
    hence thesis by A4,A3,A7,A8,PRE_TOPC:def 1;
  end;
  then
A10: (sigma T) \/ O c= the topology of T by A4,XBOOLE_1:8;
A11: the RelStr of S = the RelStr of T by YELLOW_9:def 4;
A12: sigma T = the topology of S by YELLOW_9:51;
  then sigma T is Basis of S by CANTOR_1:2;
  then
A13: sigma T is prebasis of S by YELLOW_9:27;
A14: the carrier of S in sigma T by A12,PRE_TOPC:def 1;
  K c= O
  proof
    let a be object;
    assume a in K;
    then consider x being Element of R such that
A15: a = (uparrow x)`;
    reconsider y = x as Element of T by A1;
    a = [#]T\uparrow y by A1,A15,WAYBEL_0:13;
    hence thesis by A11,A14;
  end;
  then
A16: (sigma T) \/ K c= (sigma T) \/ O by XBOOLE_1:9;
  T is TopAugmentation of T by YELLOW_9:44;
  then T is Refinement of S,R by Th29;
  hence thesis by A13,A1,A11,Th23,A16,A10,Th22;
end;
