
theorem Th51:
  for L being up-complete LATTICE holds L is meet-continuous iff
for D1, D2 being non empty directed Subset of L holds (sup D1) "/\" (sup D2) =
  sup (D1 "/\" D2)
proof
  let L be up-complete LATTICE;
  hereby
    assume L is meet-continuous;
    then for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1
    "/\" I2) by Th50;
    hence
    for D1, D2 be directed non empty Subset of L holds (sup D1) "/\" (sup
    D2) = sup (D1 "/\" D2) by Th40;
  end;
  assume for D1, D2 being non empty directed Subset of L holds (sup D1) "/\"
  (sup D2) = sup (D1 "/\" D2);
  hence L is up-complete & L is satisfying_MC by Th44;
end;
