(as in the second part of the lemma) then we also have that $|R|\le 16nk/r$.

The first property follows from the fact that each $v$ is in at least $k$ $r$-rich lines and that each $R_L$ with $v \in L$ has at least one $r$-tuple containing $v$. The second property follows from the fact that each pair $u,v$ can belong to at most one $r$-rich line $L$ and that each $R_L$ can contain at most two $r$-tuples with both $u$ and $v$. The fact that the $r$-tuple of point $u_{i_1},\ldots,u_{i_r}$ is linearly dependent follows from Claim \[cla-veronese-line\]. The fact that all the coefficients $\alpha_j$ are non zero holds since no proper subset of that $r$-tuple is linearly dependent (by the ‘moreover’ part of Claim \[cla-veronese-line\]). If each point is in at most $8k$ lines then each point is in at most $16k$ $r$-tuples (at most two on each line). This means that there could be at most $16nk/r$ tuples in $R$ since otherwise, some point would be in too many tuples.

We now construct the matrix $A$ of size $m\times n$ where $m=|R|$. For each $r$-tuple $(v_{i_1},\ldots,v_{i_r}) \in R$ we add a row to $A$ (the order of the rows 