Cone/Linear form/Coprime/Exercise

Let ${}C\subseteq \mathbb {R} ^{n}$ denote a positve rational polyhedrial cone and ${}F$ a facet of the cone. Let

$\ell \colon \mathbb {R} ^{n}\longrightarrow \mathbb {R}$

be a linear form, such that its kernel contains the facet. Suppose that the linear form is given by integers which are coprime. Show that ${}\ell$ or ${}-\ell$ is the canonical integral linear form of ${}F$.

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