Lines `L_1` and `L_2` have slopes m and n, respectively, suppose `L_1` makes twice as large angle with the horizontal (mesured counter clockwise from the positive x-axis as does `L_2` and `L_1` has 4 times the slope of `L_2`. If `L_1` is not horizontal, then the value of the proudct mn equals.

Explanation

To solve the problem, we need to find the product of the slopes \( m \) and \( n \) of the lines \( L_1 \) and \( L_2 \), given that \( L_1 \) makes twice the angle with the horizontal as \( L_2 \) and that \( m = 4n \).

### Step-by-Step Solution:

1. **Identify the Angles**:
Let the angle that line \( L_2 \) makes with the horizontal be \( \theta \). Therefore, the angle that line \( L_1 \) makes with the horizontal is \( 2\theta \).

2. **Express Slopes in Terms of Angles**:
The slopes of the lines can be expressed using the tangent function:
- For line \( L_2 \): \( n = \tan(\theta) \)
- For line \( L_1 \): \( m = \tan(2\theta) \)

3. **Use the Double Angle Formula**:
We know from trigonometry that:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Substituting for \( m \):
\[ m = \frac{2n}{1 - n^2} \]

4. **Relate Slopes**:
We are given that \( m = 4n \). Therefore, we can set up the equation:
\[ \frac{2n}{1 - n^2} = 4n \]

5. **Cross Multiply**:
To eliminate the fraction, we cross-multiply:
\[ 2n = 4n(1 - n^2) \]

6. **Expand and Rearrange**:
Expanding the right side gives:
\[ 2n = 4n - 4n^3 \]
Rearranging this leads to:
\[ 4n^3 - 2n = 0 \]

7. **Factor the Equation**:
Factoring out \( 2n \):
\[ 2n(2n^2 - 1) = 0 \]

8. **Solve for \( n \)**:
This gives us two cases:
- \( 2n = 0 \) (which we discard since \( L_1 \) is not horizontal)
- \( 2n^2 - 1 = 0 \) leading to \( n^2 = \frac{1}{2} \) or \( n = \frac{1}{\sqrt{2}} \)

9. **Find \( m \)**:
Since \( m = 4n \):
\[ m = 4 \cdot \frac{1}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]

10. **Calculate the Product \( mn \)**:
Now, we find the product \( mn \):
\[ mn = m \cdot n = (2\sqrt{2}) \cdot \left(\frac{1}{\sqrt{2}}\right) = 2 \]

### Final Answer:
The value of the product \( mn \) is \( \boxed{2} \).