**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}\$$).