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16 (The Gyrotranslation Theorem, I). Let (G, +) be a gyrogroup. 22) for all a, b, c ∈ G. Proof. 20) and a left cancellation. 23) which generalizes the obvious group identity (−a+b)+(−b+c)+(−c+d) = −a + d. 15 suggests the following definition. 17 (Gyropolygonal Gyroaddition of Adjacent Sides). Let (G, +) be a gyrogroup, and let (a, b), a, b ∈ G be a pair of two elements of G. January 14, 2008 9:33 WSPC/Book Trim Size for 9in x 6in 24 ws-book9x6 Analytic Hyperbolic Geometry (i) (ii) (iii) (iv) The value of the pair (a, b) is −a + b ∈ G.

The conversion law from gyrotriangle gyroangles α, β, γ to their corresponding gyrotriangle side gyrolengths a , b , c in a gyrotriangle ABC is shown in Fig. 13, p. 49, p. 307. It presents one of the examples of an extraordinary January 14, 2008 12 9:33 WSPC/Book Trim Size for 9in x 6in Analytic Hyperbolic Geometry unexpected hidden beauty that analytic hyperbolic geometry reveals. We thus encounter in this book remarkable interrelations between truth and beauty, which are abound in the areas of analytic hyperbolic geometry, and which share remarkable analogies with corresponding classical results in the areas of analytic Euclidean geometry.

9). 10). 121) of the left loop property followed by a left cancellation. 123), of the left loop property followed by a left cancellation. 112). 128) gyr[gyr[a, −b]b, a] = gyr[a, −b] for all a, b ∈ G. Proof. 131) for all a, b ∈ G. 126). 126). 127). 128) is equivalent to the first one. 128) follows from the first (third) by replacing a by −a (or, alternatively, by replacing b by −b). January 14, 2008 9:33 WSPC/Book Trim Size for 9in x 6in 42 ws-book9x6 Analytic Hyperbolic Geometry We are now in a position to find that the left gyroassociative law and the left loop property of gyrogroups have right counterparts.