
theorem Th50:
  for L being up-complete LATTICE holds L is meet-continuous iff
  for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2)
proof
  let L be up-complete LATTICE;
  hereby
    assume L is meet-continuous;
    then SupMap L is meet-preserving join-preserving;
    hence for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1
    "/\" I2) by Th38;
  end;
  assume
  for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2);
  then
  for D1, D2 be directed non empty Subset of L holds (sup D1) "/\" (sup D2
  ) = sup (D1 "/\" D2) by Th40;
  hence L is up-complete & L is satisfying_MC by Th44;
end;
