We prove that for any properly two-colored arrangement of lines in the Euclidean plane having, say, r red and g green regions with
r
⩾
g
r \geqslant g
, the inequality
\[
r
⩽
2
g
−
2
−
∑
P
(
λ
(
P
)
−
2
)
r \leqslant 2g - 2 - \sum \limits _P {(\lambda (P) - 2)}
\]
holds, where for each point P of intersection of the lines,
λ
(
P
)
\lambda (P)
is the number of lines of the arrangement that contain P. This strengthens recent results of Simmons and Grünbaum.