Shortest-independent-vector basis lemma
= Shortest-independent-vector basis lemma
In a rank-two <Euclidean lattice>, a shortest nonzero vector $v$ and a shortest vector $w\notin\mathbb Zv$ form a basis. The shortest vector is primitive. If the vertical coordinate of $w$ has index $m\geq2$ in the projection lattice, a vector with smaller positive vertical coordinate can be reduced horizontally to length squared at most $(|v|^2+|w|^2)/4<|w|^2$, a contradiction. This lemma need not hold in higher rank.