
theorem Th53:
  for L being up-complete Semilattice holds L is meet-continuous
  iff inf_op L is directed-sups-preserving
proof
  let L be up-complete Semilattice;
  hereby
    assume L is meet-continuous;
    then
    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;
    hence inf_op L is directed-sups-preserving by Th46;
  end;
  assume inf_op L is directed-sups-preserving;
  then
  for D1, D2 being non empty directed Subset of L holds (sup D1) "/\" (sup
  D2) = sup (D1 "/\" D2) by Th43;
  hence L is up-complete & L is satisfying_MC by Th44;
end;
