  For an arrangement of $n$ pseudolines in the real projective plane let us
denote by $t_i$ the number of vertices incident to $i$ lines. We obtain a
linear on $t_i$ inequality similar to the Hirzebruch one, but with an
elementary proof. We present an algorithm for producing lower bounds of the
number of regions basing on linear on $t_i$ inequalities like the
above-mentioned. Lower bounds arise in connection with Martinov theorem on the
set of all possible numbers of regions and we show how the new bounds may be
applied in it.
