Into how many regions do $n$ lines divide the plane if at most $n-k$ of them are concurrent?

A number of connective components of the real projective plane, disjoint with the family of $n\geq 2$ distinct lines is estimated provided at most $n-k$ lines are concurrent. If $n\geq\frac{k^2+k}2+3$, then the number of regions is at least $(k+1)(n-k)$. Thus, a new proof of Martinov's theorem is obtained. This theorem determines all pairs of integers $(n,f)$ such that there is an arrangement of $n$ lines dividing the projective plane into $f$ regions.