
theorem Th28:
  for N being meet-continuous LATTICE, X, Y being upper Subset of
  N holds (X^0) \/ (Y^0) = (X \/ Y)^0
proof
  let N be meet-continuous LATTICE, X, Y be upper Subset of N;
  thus (X^0) \/ (Y^0) c= (X \/ Y)^0 by Th27;
  assume not (X \/ Y)^0 c= (X^0) \/ (Y^0);
  then consider s being object such that
A1: s in (X \/ Y)^0 and
A2: not s in (X^0) \/ (Y^0);
A3: not s in X^0 by A2,XBOOLE_0:def 3;
A4: not s in Y^0 by A2,XBOOLE_0:def 3;
  reconsider s as Element of N by A1;
  consider D being non empty directed Subset of N such that
A5: s <= sup D and
A6: X misses D by A3;
  consider E being non empty directed Subset of N such that
A7: s <= sup E and
A8: Y misses E by A4;
  s "/\" s = s by YELLOW_0:25;
  then s <= (sup D) "/\" (sup E) by A5,A7,YELLOW_3:2;
  then
A9: s <= sup (D "/\" E) by WAYBEL_2:51;
  ex xy being Element of N st s = xy & for D being non
  empty directed Subset of N st xy <= sup D holds (X \/ Y) meets D by A1;
  then (X \/ Y) meets (D"/\"E) by A9;
  then
A10: (X \/ Y) /\ (D"/\"E) <> {};
  X misses (D "/\" E) by A6,YELLOW12:21;
  then
A11: X /\ (D "/\" E) = {};
  Y misses (D "/\" E) by A8,YELLOW12:21;
  then X /\ (D"/\"E) \/ Y /\ (D"/\"E) = {} by A11;
  hence contradiction by A10,XBOOLE_1:23;
end;
