
theorem
  for L being non empty Poset holds L is completely-distributive iff
    L opp is completely-distributive
proof
  let L be non empty Poset;
  thus L is completely-distributive implies L opp is completely-distributive
  proof
    assume
A1: L is completely-distributive;
    hence L opp is complete;
    let J be non empty set, K be non-empty ManySortedSet of J;
    let F be DoubleIndexedSet of K, L opp;
    reconsider G = F as DoubleIndexedSet of K, L;
    thus Inf Sups F = \\/(Sups F, L) by A1,Th49
      .= Sup Infs G by A1,Th50
      .= Inf Sups Frege G by A1,Th48
      .= //\(Infs Frege F, L) by A1,Th50
      .= Sup Infs Frege F by A1,Th49;
  end;
  assume
A2: L opp is completely-distributive;
  then
A3: L is complete non empty Poset by Th17;
  thus L is complete by A2,Th17;
  let J be non empty set, K be non-empty ManySortedSet of J;
  let F be DoubleIndexedSet of K, L;
  reconsider G = F as DoubleIndexedSet of K, L opp;
  thus Inf Sups F = \\/(Sups F, L opp) by A3,Th49
    .= Sup Infs G by A3,Th50
    .= Inf Sups Frege G by A2,Th48
    .= //\(Infs Frege F, L opp) by A3,Th50
    .= Sup Infs Frege F by A3,Th49;
end;
