While retreating, Fencer X parries Fencer Ys attack and ripostes, touching Y on the valid surface, which is registered on the machine. X retreats then falls; Ys remise arrives on the valid surface, which is registered on the machine?

Answer: Award a touch for X

Step-by-step explanation:

This is not a Mathematical question, it's a fencing question. It just requires the knowledge of the fencing sport.

Knowing the rules of the game, would let one know that fencer X landed the first legal touch on the valid surface of fencer Y.

Fencer Y's touch on Fencer X's valid surface arrives when fencer X already tripped while retreating and fell.

QED!