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;

theorem Th32:
  I is max-ideal iff I.: is being_ultrafilter
proof
  thus I is max-ideal implies I.: is being_ultrafilter
  proof
    assume that
A1: I <> carr(L) and
A2: for J st I c= J & J <> carr(L) holds I = J;
    thus I.: <> carr(L.:) by A1;
    let F be Filter of L.:;
    .:F = F;
    hence thesis by A2;
  end;
  assume that
A3: I.: <> carr(L.:) and
A4: for F being Filter of L.: st I.: c= F & F <> carr(L.:) holds I.: = F;
  thus I <> carr(L) by A3;
  let J be Ideal of L;
  J.: = J;
  hence thesis by A4;
end;
