Equivalences
In other treatments of elementary geometry, using different sets of axioms, Pasch's axiom can be proved as a theorem; it is a consequence of the plane separation axiom when that is taken as one of the axioms. Hilbert uses Pasch's axiom in his axiomatic treatment of Euclidean geometry. Given the remaining axioms in Hilbert's system, it can be shown that Pasch's axiom is logically equivalent to the plane separation axiom.
Read more about this topic: Pasch's Axiom