We defined all the h-notions needed in the formulation of the axioms I-IV and h-V It remains to show that all these axioms hold; this will be done by the end of this chapter.
Once we are done with the proofs, we get that the model provides an example of a neutral plane; in particular, Exercise 12.1.5 can be proved the same way as Theorem 5.3.1.
Most importantly we will prove the “if”-part of Theorem 11.5.2.
Indeed, any statement in hyperbolic geometry can be restated in the Euclidean plane using the introduced h-notions. Therefore, if the system of axioms I-IV, and h-V leads to a contradiction, then so does the system axioms I-V.