By Amnon Neeman

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112) becomes a + b = −gyr[a, b](−b − a) . 111) complete the proof. 26 in terms of the gyrosemidirect product group. 14, p. 12. 27. Let (G, +) be a gyrogroup. 118) gyr[a, b] = gyr[−(a + b), a] . 9. BASIC GYRATION PROPERTIES 27 Proof. 40). 41). 115) of the left loop property followed by a left cancellation. 115), gyr[a, b] = gyr−1 [−b, −a] = gyr[b, −b − a] . 117), of the left loop property followed by a left cancellation. 106). (The Gyration Inversion Law; The Gyration Even Property). 28. 123) gyr[gyr[a, −b]b, a] = gyr[a, −b] for all a, b ∈ G.

9, p. 33. ✷ Let (G, ⊕) be a gyrogroup. Then a⊕{( a⊕b)⊕a} = b for all a, b ∈ G. 10. AN ADVANCED GYROGROUP EQUATION 31 Proof. 135) === b⊕gyr[b, a]a (4) === b a. 135) follows. (1) Follows from the left gyroassociative law. (2) Follows from (1) by a left cancellation, and by a left loop followed by a left cancellation. (3) Follows from (2) by a right loop, that is, an application of the right loop property to (3) gives (2). (4) Follows from (3) by Def. 9, p. 7, of the gyrogroup cooperation . 34 below an advanced gyrogroup equation and its unique solution.

32. −(a for any a, b∈G. 132) 30 CHAPTER 1. GYROGROUPS Proof. 133) [a, −b]{−(−gyr[a, −b]b − a)} (6) === −(−b − gyr−1 [a, −b]a) (7) === −{−b − gyr[b, −a]a} (8) === −{(−b) (−a)} . 132). 133) follows. (1) Follows from Def. 9, p. 7, of the gyrogroup cooperation . 105). 13(12) applied to the term {. } in (2). 13(12) applied to b, that is, gyr[a, −b]b = −gyr[a, −b](−b). 27. (6) Follows from (5) by distributing the gyroautomorphism gyr−1 [a, −b] over each of the two terms in {. }. 106). (8) Follows from (7) by Def.