Identity of Points

Theorem

Two points share the same position if and only if they are the same points.

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.

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.

$\blacksquare$

Also see

• Identity of Equidistance, an analogue of Identity of Points in the context of Tarski's Geometry.