A very short algebraic proof of the Farkas Lemma

We present a very short algebraic proof of a generalisation of the Farkas Lemma: we set it in a vector space of finite or infinite dimension over a linearly ordered (possibly skew) field; the non‐positivity of a finite homogeneous system of linear inequalities implies the non‐positivity of a linear mapping whose image space is another linearly ordered vector space. In conclusion, we briefly discuss other algebraic proofs of the result, its special cases and related results.