Here is an explanation of why the spacetime interval $c^2t^2-x^2-y^2-z^2$ is invariant under Lorentz transformation in Wikipedia. Link of the proof. I kind of understand it, but the explanation does not mention anything specific about the speed of light $c$. No properties of $c$ are used explicitly. I am confused by this because the proof still seems to work even if I replace $c$ with any other constant.
Which steps in the proof use the property of $c$ ? If I am not clear enough, Why the proof become invalid if I replace $c$ with another constant in the proof?