Prove that there exists a $k_0\in\mathbb{N}$ such that for every $k\in\mathbb{N},k>k_0$, there exists a finite number of lines in the plane not all parallel to one of them, that divide the plane exactly in $k$ regions. Find $k_0$.
1. **Understanding the problem**: We need to prove that there exists a natural number \( k_0 \) such that for every \( k \in \mathbb{N} \) with \( k > k_0 \), there exists a finite number of lines in the plane, not all parallel to one another, that divide the plane into exactly \( k \) regions. We also need to find the smallest such \( k_0 \).

2. **Analyzing the number of regions formed by lines**: 
   - For \( n \) lines in general position (no two lines are parallel and no three lines are concurrent), the maximum number of regions \( R(n) \) formed is given by:
     \[
     R(n) = \frac{n(n+1)}{2} + 1
     \]
   - For example:
     - 1 line divides the plane into 2 regions.
     - 2 lines (if not parallel) divide the plane into 4 regions.
     - 3 lines (if no two are parallel and no three are concurrent) divide the plane into 7 regions.
     - 4 lines (in general position) divide the plane into 11 regions.
     - 5 lines (in general position) divide the plane into 16 regions.

3. **Special cases with fewer regions**:
   - If some lines are parallel or concurrent, the number of regions decreases.
   - For 5 lines:
     - If one pair is parallel, we get 15 regions.
     - If two pairs are parallel, we get 14 regions.
     - If one triple is parallel, we get 13 regions.
     - If two pairs are parallel and the fifth line intersects at two points, we get 12 regions.

4. **Generalizing for \( N \geq 12 \)**:
   - For \( N \geq 12 \), we can write \( N = 5k + m \) where \( m \in \{12, 13, 14, 15, 16\} \).
   - We can start with an arrangement of 5 lines that gives \( m \) regions.
   - By adding \( k \) lines parallel to one of the existing lines (which is not parallel to any other line), we can increase the number of regions by \( k \), thus achieving \( N \) regions.

5. **Finding the smallest \( k_0 \)**:
   - We need to check the smallest number of regions that can be formed with fewer lines:
     - With 4 lines, we can get up to 11 regions.
     - With 3 lines, we can get up to 7 regions.
     - With 2 lines, we can get up to 4 regions.
     - With 1 line, we get 2 regions.
   - We observe that we cannot get exactly 5 regions without having all lines parallel.
   - Therefore, the smallest \( k_0 \) such that for every \( k > k_0 \), there exists a finite number of lines not all parallel that divide the plane into exactly \( k \) regions is \( k_0 = 5 \).

The final answer is \( \boxed{ k_0 = 5 } \).