spacetime - Special Relativity - Events "Coincide" is NOT a relative concept, Why?

Consider 1-D space. Let S and S' be two inertial reference frames. Let A and B be two events.

Co-ordinates of A and B under S are A = (xA,tA) and B = (xB,tB).

When we say events coincide - it simply means they have same space-time co-ordinates.

i.e. if (xA = xB) and (tA = tB), then w.r.t S, events A and B coincide.

Let me state a theorem: If A and B coincide in S, then they will the coincide in S' (hence in every and any IRF i.e. two events being coincident is NOT a relative concept)

Q1 - Why is this theorem? Is there a deeper assumption and understanding regarding space-time behind this concept? (I'm not looking for an answer basis Lorentz Transformation - but a more physical / maybe more basic argument). Or is this just an assumption of Special Relativity?

Q2 - If 2 balls A and B collide - they will collide in every IRF. How can I derive this basis above theorem? i.e. how can I "precisely" express collision of 2 balls as two events which coincide?

(I'm asking the above question to better understand space-time, events etc at a little conceptual level and I'm having difficulty in understanding them, Thanks for your help)

Answer

It's a lot simpler than you think. Suppose an event has coordinates $x$ in some reference frame, where $x$ contains both space and time coordinates within it. To get the coordinates $x'$ of the same event in some other reference frame, you apply some function,

$$x' \equiv f(x).$$ This works in nonrelativistic physics, special relativity, and even general relativity. In special relativity the function is called a Lorentz transformation. The key (essentially only) assumption here is that the location of an event in spacetime is completely specified by its coordinates.

If two events $A$ and $B$ coincide, their coordinates are the same, $x_A = x_B$. You want a proof of the "theorem" that in any other reference frame, $x_A' = x_B'$. Now hold on to your seat, because this profound result can be beautifully proven in airtight, perfectly rigorous formal mathematics as:

$$x_A' = f(x_A) = f(x_B) = x_B'.$$ That's it.