This is a basic (seemingly) basic lemma on rational polyhedral cones that is sometimes used to work with toric varieties and is generally "left to the reader". Unfortunately, I could neither prove it myself nor find complete proof in the literature. So I'm looking for either proof or an appropriate reference.

Lemma.Let $ V $ bean $ mathbb {R} $-vector of finite dimensional space, and let $ N $ be a $ mathbb {Z} $-structure on $ V $ (that is, 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 $.

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

**Note 1** Try to show (iii)$ Rightarrow $(i) in a manner similar to (ii)$ Rightarrow $(i) asks 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 monoid face $ sigma cap N $and likewise for (ii).