Introducing a lemma is like changing your foliation
3.6 Euclidean Plane Geometry
The axioms of Euclidean plane geometry are the axioms of predicate logic together with the thirteen axioms below. This formulation of the axioms is taken from Tarski's paper "What is elementary geometry?". As is mentioned in his paper, these axioms are complete (syntactically) and consistent. Tarski claims in his book A Decision Method for Elementary Algebra and Geometry that these axioms describe essentially the same geometry as Hilbert (who first formalized Euclid's axioms).
The sets and for Euclidean plane geometry are defined as follows.