1. The fact is that all isometries are invertible, so we do not, in reality, lose anything by making this requirement. However, it will not be worth the effort to prove this. We will be able to accomplish anything we want to accomplish using isometries with only translations, reflections, rotations, and compositions thereof, and we will prove that a trnslation is invertible in Theorem 4.11, and in Theorem 4.9, we will prove that if you compose a reflection with itself, you will get the identity function, making a reflection equal to its own inverse. Rotations will have to wait until chapter 5, since they depend upon angles, but when we get to them, we will prove that a rotation is the composition of two reflections, and since we prove, in Theorem 4.4, that the composition of two isometries is an isometry, it will follow that a rotation is also an isometry.

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Analytic Foundations of Geometry

R. S. Wilson