theorem
  LowerCone([#](A)) = {}
proof
  thus LowerCone([#](A)) c= {}
  proof
    let x be object;
    assume x in LowerCone([#](A));
    then ex a st x = a & for a2 st a2 in [#](A) holds a < a2;
    hence thesis;
  end;
  thus thesis;
end;
