Identity of Points
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Two points share the same position if and only if they are the same points.
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.
- $P_1 = P_2 \dashv \vdash a = b$
Hence the result.
- Identity of Equidistance, an analogue of Identity of Points in the context of Tarski's Geometry.