Identity of Points

Theorem
Two points share the same position iff they are the same point.

Proof
Let $a$ be a point with position $P_1$.

Let $b$ be a point with position $P_2$.

By hypothesis, $P_1 = P_2$.

By Leibniz's Law, two objects are the same object if and only if they share every property in common.

But by the definition of point, the only property possessed by a point is position.

We have:


 * $P_1 = P_2 \dashv \vdash a = b$

Hence the result.