QUESTION: Does exist a proof of statement $ P = L - I + 1$, where $P$ stands for number of polygons, $L$ for number of line segments, $I$ for number of intersection points, in any arrangement made by finite number of lines in $2D$ space?
EXAMPLE: On Arrangement of lines on a plane, I=10, L=15 and P=6 obviously there is $10$ intersection points (red), $15$ line segments (blue), and $6$ polygons (made by these line segments), which satisfies the statement because $ 6 = 15 - 10 + 1$.
ADDITIONAL INFORMATIONS: In general, for $n$ lines, $n\geq3$ we have $$ I = \binom{n}{2} - I_l $$ and $$ L = n(n-2) - L_l $$ where $I_l,L_l$ stand for intersection points and line segments lost due to parallel or concurrent lines for any configuration in a plane.
If all $n$ lines are parallel, statement is false, because $0 \neq 0 - 0 + 1$.
BACK TO QUESTION: Is this statement $ P = L - I + 1$ true? For arrangement with which conditions? Is there some proof?
BONUS: Is there a similar statement in higher dimensions?