Does every Partially Ordered relation and its dual have the same number of topological orderings?

Question

Given the Hasse Diagram of a Partially Ordered Relation, is it the case that both the POSET itself and its dual POSET have the same number of topological orderings? I have tried a few examples, and although it does seem to be the case, I need a formal way of proving the same. So please help me with this.