
theorem
  for S being compact Hausdorff TopLattice st for x being Element of S
  holds x"/\" is continuous holds S is bounded
proof
  let S be compact Hausdorff TopLattice such that
A1: for x being Element of S holds x"/\" is continuous;
  thus S is lower-bounded
  proof
    reconsider x = inf (S opp+id) as Element of S;
    take x;
A2: rng the mapping of S opp+id = rng id S by Def6
      .= the carrier of S by RELAT_1:45;
    then
A3: x = "/\"(the carrier of S,S) by YELLOW_2:def 6;
    ex_inf_of S opp+id by A1,Th39;
    then ex_inf_of the carrier of S, S by A2;
    hence thesis by A3,YELLOW_0:31;
  end;
  reconsider x = sup (S+id) as Element of S;
  take x;
A4: rng the mapping of S+id = rng id S by Def5
    .= the carrier of S by RELAT_1:45;
  then
A5: x = "\/"(the carrier of S,S) by YELLOW_2:def 5;
  ex_sup_of S+id by A1,Th38;
  then ex_sup_of the carrier of S, S by A4;
  hence thesis by A5,YELLOW_0:30;
end;
