
theorem Th49:
  for L being up-complete LATTICE holds L is meet-continuous iff
  SupMap L is meet-preserving join-preserving
proof
  let L be up-complete LATTICE;
  hereby
    assume
A1: L is meet-continuous;
    for D1, D2 being non empty directed Subset of L holds (sup D1) "/\" (
    sup D2) = sup (D1 "/\" D2)
    proof
      let D1, D2 be non empty directed Subset of L;
      for x being Element of L, E being non empty directed Subset of L st
      x <= sup E holds x <= sup ({x} "/\" E) by A1,Th45;
      then inf_op L is directed-sups-preserving by Th46;
      hence thesis by Th43;
    end;
    then for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1
    "/\" I2);
    hence SupMap L is meet-preserving join-preserving by Th39;
  end;
  assume
A2: SupMap L is meet-preserving join-preserving;
  thus L is up-complete;
  for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\"
  I2 ) by A2,Th38;
  then
  for D1, D2 be non empty directed Subset of L holds (sup D1) "/\" (sup D2
  ) = sup (D1 "/\" D2) by Th40;
  hence thesis by Th44;
end;
