reserve X,Y,x,y for set;
reserve A for non empty Poset;
reserve a,a1,a2,a3,b,c for Element of A;
reserve S,T for Subset of A;

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