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.

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

This Sketch 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!