
theorem
  for T being Lawson complete TopLattice holds {W\uparrow F where W,F
  is Subset of T: W in sigma T & F is finite} is Basis of T
proof
  let T be Lawson complete TopLattice;
  set R = the lower correct TopAugmentation of T;
  reconsider B2 = {(uparrow F)` where F is Subset of R: F is finite} as Basis
  of R by Th7;
  set Z = {W\uparrow F where W,F is Subset of T: W in sigma T & F is finite};
  set S = the Scott TopAugmentation of T;
A1: the topology of S = sigma T by YELLOW_9:51;
  then reconsider B1 = sigma T as Basis of S by CANTOR_1:2;
A2: the RelStr of R = the RelStr of T by YELLOW_9:def 4;
  B1 c= Z
  proof
    set F9 = {}R;
    reconsider G = F9 as Subset of T by A2;
    let x be object;
    assume
A3: x in B1;
    then reconsider x9 = x as Subset of T;
    uparrow G = uparrow F9;
    then x9\uparrow G = x9;
    hence thesis by A3;
  end;
  then
A4: B1 \/ Z = Z by XBOOLE_1:12;
A5: INTERSECTION(B1,B2) = Z
  proof
    hereby
      let x be object;
      assume x in INTERSECTION(B1,B2);
      then consider y,z being set such that
A6:   y in B1 and
A7:   z in B2 and
A8:   x = y /\ z by SETFAM_1:def 5;
      reconsider y as Subset of T by A6;
A9:   [#]T /\ y = y by XBOOLE_1:28;
      consider F being Subset of R such that
A10:  z = (uparrow F)` and
A11:  F is finite by A7;
      reconsider G = F as Subset of T by A2;
      z = (uparrow G)` by A2,A10,WAYBEL_0:13;
      then x = y\uparrow G by A9,A8,XBOOLE_1:49;
      hence x in Z by A6,A11;
    end;
    let x be object;
    assume x in Z;
    then consider W, F being Subset of T such that
A12: x = W\uparrow F and
A13: W in sigma T and
A14: F is finite;
    W /\ [#]T = W by XBOOLE_1:28;
    then
A15: x = W /\ ([#]T\uparrow F) by A12,XBOOLE_1:49;
    reconsider G = F as Subset of R by A2;
A16: (uparrow G)` in B2 by A14;
    (uparrow F)` = (uparrow G)` by A2,WAYBEL_0:13;
    hence thesis by A16,A13,A15,SETFAM_1:def 5;
  end;
A17: the RelStr of S = the RelStr of T by YELLOW_9:def 4;
  B2 c= Z
  proof
    let x be object;
    assume x in B2;
    then consider F being Subset of R such that
A18: x = (uparrow F)` and
A19: F is finite;
A20: the carrier of S in B1 by A1,PRE_TOPC:def 1;
    reconsider G = F as Subset of T by A2;
    uparrow F = uparrow G by A2,WAYBEL_0:13;
    hence thesis by A17,A20,A2,A18,A19;
  end;
  then
A21: B2 \/ Z = Z by XBOOLE_1:12;
  T is TopAugmentation of T by YELLOW_9:44;
  then T is Refinement of S,R by Th29;
  then B1 \/ B2 \/ INTERSECTION(B1,B2) is Basis of T by YELLOW_9:59;
  hence thesis by A4,A5,A21,XBOOLE_1:4;
end;
