reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;
reserve I,J for Ideal of L,
  F for Filter of L;
reserve D for non empty Subset of L,
  D9 for non empty Subset of L.:;
reserve D1,D2 for non empty Subset of L,
  D19,D29 for non empty Subset of L.:;

theorem Th43:
  I is prime iff I.: is prime
proof
  thus I is prime implies I.: is prime
  proof
    assume
A1: p "/\" q in I iff p in I or q in I;
    let p9,q9;
    p9"\/"q9 = ( .:p9)"/\"( .:q9);
    hence thesis by A1;
  end;
  assume
A2: p9"\/"q9 in I.: iff p9 in I.: or q9 in I.:;
  let p,q;
  (p.:)"\/"(q.:) = p"/\"q;
  hence thesis by A2;
end;
