
theorem
  for R being complete connected LATTICE,
  T being Scott TopAugmentation of R, S being Subset of T holds
  S is open iff S = the carrier of T or S in the set of all (downarrow x)`
  where x is Element of T
proof
  let R be complete connected LATTICE,
  T be Scott TopAugmentation of R, S be Subset of T;
  set DD = the set of all (downarrow x)` where x is Element of T;
  thus S is open implies S = the carrier of T or S in DD
  proof
    assume
A1: S is open;
    assume that
A2: S <> the carrier of T and
A3: not S in DD;
A4: [#]T\S <> {} by A2,PRE_TOPC:4;
A5: the RelStr of T = the RelStr of R by YELLOW_9:def 4;
    then reconsider K = S` as Subset of R;
    reconsider D = K as non empty directed Subset of T by A4,A5,WAYBEL_0:3;
A6: D misses S by SUBSET_1:23;
    reconsider x = sup D as Element of T;
    S` = downarrow x
    proof
      thus S` c= downarrow x
      proof
        let a be object;
        assume
A7:     a in S`;
        then reconsider A = a as Element of T;
        x is_>=_than S` by YELLOW_0:32;
        then A <= x by A7;
        hence thesis by WAYBEL_0:17;
      end;
      let a be object;
      assume
A8:   a in downarrow x;
      then reconsider A = a as Element of T;
A9:   A <= x by A8,WAYBEL_0:17;
      not x in S by A1,A6,WAYBEL11:def 1;
      then not A in S by A1,A9,WAYBEL_0:def 20;
      hence thesis by XBOOLE_0:def 5;
    end;
    then (downarrow x)` = S;
    hence contradiction by A3;
  end;
  assume
A10: S = the carrier of T or S in DD;
  per cases by A10;
  suppose
A11: S = the carrier of T;
A12: the RelStr of T = the RelStr of R by YELLOW_9:def 4;
    then S = [#]R by A11;
    then
A13: S is inaccessible by A12,YELLOW_9:47;
    S is upper
    by A11;
    hence thesis by A13;
  end;
  suppose S in DD;
    then ex x being Element of T st ( S = (downarrow x)`);
    hence thesis by Th19;
  end;
end;
