**KEP1:** Recall that the [principal downset](https://en.wikipedia.org/wiki/Upper_set) for an element \\(z\\) is defined as \\(\operatorname{\downarrow}(z) = \\{y \in P : y \le z\\}\\). Assume that \\(P\\) has a maximum, \\(\top\\), and let \\(x \in P\\). Then \\(x \le \top\\) by definition, so \\(x \in \operatorname{\downarrow}(\top)\\). Therefore \\(P \subseteq \operatorname{\downarrow}(\top)\\). Since we also have \\(\operatorname{\downarrow}(\top) \subseteq P\\) by definition, we know \\(P = \operatorname{\downarrow}(\top)\\).

**KEP2:** By duality (upsets in \\(P\\) are downsets in \\(P^{op}\\)).

**KEP2:** By duality (upsets in \\(P\\) are downsets in \\(P^{op}\\)).