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By Pierre de la Harpe

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So, point E is one which depends rationally upon A, B, C. It is easy to prove that all points, except for those lying on the three sides AB, AC and BC of the triangle ABC, which depend rationally upon A, B, C can be divided into the interior and the exterior two parts - called the rational interior part and rational exterior part - of triangle ABC which have those properties concerning the interior and exterior of triangles in usual geometry. Although for all points in a plane the corresponding Pasch's axiom may not necessarily be satisfied or may even become meaningless, we can still prove that all points which depend rationally upon A, B, C satisfy the so-called Pasch's axiom of the triangle ABC.

By Axiom H IV again, these two parallels are not parallel but necessarily meet at a point D; then AB DC is a parallelogram. Instead of the axioms of order, we introduce the following. Axiom of infinity I. Let Ao, Al be any two points on a line I and construct an arbitrary OAoAIBe. 1 Through B draw a line BA2 /I CAl, meeting I at point A2, and then a line BA3 II CA2, meeting I at point A3, and so on analogously. Similarly, through C draw a line CA-I II BAo, meeting I at point A-I, and then a line CA-2 II BA_I, meeting l at point A-2, and so on analogously.

Or This is what we needed to prove. Similarly, for proving (a + b)' = a' + b', in the case AB' I A' B the direct geometric proof can also be transformed into a simple proof by using properties of the number system. , the direct geometric proof starting from geometric axioms and the algebraic proof based on properties of the number system, one may observe that for the former the introduction of auxiliary lines is not simple and the proof steps follow no rules, whereas for the latter the deduction rules can be more easily sought from the calculations, which thus provides the possibility for mechanical proving.

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