Proposition 13

In a lattice, the intersection of two intervals is an interval. Consider for example the intersection of two closed intervals $[a, b]$ and $[c, d]$, and let $\alpha = \sup (a, c)$, $\beta = \inf (b, d)$. If both $a \leqslant x \leqslant b$ and $c \leqslant x \leqslant d$, then we have $\alpha \leqslant x \leqslant \beta$, and conversely; if $\alpha \not\leqslant \beta$, the intersection of $[a, b]$ and $[c, d]$ is empty; if $\alpha \leqslant \beta$, this intersection is $[\alpha, \beta]$. We leave it to the reader to carry through the proof for the other cases.