Let $\mathbb{P}_1$ and $\mathbb{P}_2$ be posets. Consider the product order $\mathbb{P}_1 \times \mathbb{P}_2$. Prove that $\langle a, b \rangle$ covers $\langle c, d \rangle$ in the product order if and only if
($a = c$ and $b$ covers $d$) or ($a$ covers $c$ and $b = d$)
If $\langle a,b \rangle \succ \langle c, d \rangle$, then $a \geq c$ and $b \geq d$. Suppose both $a > c$ and $b > d$ happen.
Then $\langle a,b \rangle > \langle a, d \rangle > \langle c , d \rangle$.
So it must be $a=c$ or $b=d$.