
theorem
  for L being up-complete LATTICE holds L is meet-continuous iff for x
being Element of L, D being non empty directed Subset of L st x <= sup D holds
  x = sup ({x} "/\" D)
proof
  let L be up-complete LATTICE;
  thus L is meet-continuous implies for x being Element of L, D being non
  empty directed Subset of L st x <= sup D holds x = sup ({x} "/\" D) by Th45;
  assume for x being Element of L, D being non empty directed Subset of L st
  x <= sup D holds x = sup ({x} "/\" D);
  then
  for x being Element of L, D being non empty directed Subset of L st x <=
  sup D holds x <= sup ({x} "/\" D);
  then inf_op L is directed-sups-preserving by Th46;
  then
  for D1, D2 being non empty directed Subset of L holds (sup D1) "/\" (sup
  D2) = sup (D1 "/\" D2) by Th43;
  hence thesis by Th51;
end;
