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Mathematics LibreTexts

12.2: Plan of the proof

  • Page ID
    23655
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    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.