I have $J\cdot L$ hyperplanes in $\mathbb{R}^{J-1}$ and want to prove that there cannot be more than $L$ points where $J$ hyperplanes intersect simultaneously (aka. concurrencies).

I suspect that the problem is straightforward once formulated in a clean matrix representation.

This [Sketch][1] shows the problem for $J=3$ (triangle) and $L=4$, for a total of $12$ lines. So here, the result would be that there cannot be more than four triple-intersections of the lines.

If it matters for the proof: 
- From context, I know that there are *at least* $L$ intersections, and that no lines are parallel (as in the sketch, but generalized to higher dimensions). But that should not be required since I am only missing the upper bound, which I suspect to be $L$ as well.
- As in the sketch, I can easily split the $J\cdot L$ lines into $J$ chunks of $L$ lines each. Also in higher dimensions.

Any help would be greatly appreciated!

  [1]: https://i.sstatic.net/gcoqK.png