This is about a (seemingly) basic lemma about rational polyhedral cones that is sometimes used when working with toric varieties and is usually "left to the reader". Unfortunately, I could neither prove it myself nor find a complete proof in the literature. So, I am looking for either a proof or a proper reference.

Lemma.Let $V$ be an $\mathbb{R}$-vector space of finite dimension, and let $N$ be a $\mathbb{Z}$-structure on $V$ (i.e., a free abelian group with $N\otimes_{\mathbb{Z}}\mathbb{R}=V$). Let $\sigma$ and $\tau$ be $N$-rational polyhedral cones in $V$, and suppose that $\tau$ is asubsetof $\sigma$. Then, the following statements are equivalent:(i) $\tau$ is a face of $\sigma$;

(ii) If $x,y\in\sigma$ with $x+y\in\tau$, then $x,y\in\tau$;

(iii) If $x,y\in\sigma\cap N$ with $x+y\in\tau$, then $x,y\in\tau$.

Showing that (i) and (ii) are equivalent and imply (iii) is clear. My problem is to show that (iii) implies (i) or (ii), i.e., that it suffices to consider only rational points.

**Note 1.** Trying to show (iii)$\Rightarrow$(i) analogously to (ii)$\Rightarrow$(i) leads to the question whether $(\sigma-\tau)\cap N=(\sigma\cap N)-(\tau\cap N)$, which has a negative answer in general.

**Note 2.** Condition (iii) is sometimes expressed by saying that the monoid $\tau\cap N$ is a face of the monoid $\sigma\cap N$, and similarly for (ii).