By Lucien Guillou, Alexis Marin

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**Example text**

53). 16, according to which the gyration product applied to a in (3) is trivial. (4) Follows from (3) by the left gyroassociative law. Indeed, an application of the left gyroassociative law to (4) results in (3). (5) Follows from (4) since gyr[b, a]a is the unique inverse of gyr[b, a]a. (6) Follows from (5) since 0 is the unique identity element of the gyrogroup (G, ⊕). 6. THE BASIC CANCELLATION LAWS OF GYROGROUPS 19 Formalizing the results of this section, we have the following theorem. (The Two Basic Equations Theorem).

8) holds for all b, c∈G. 2, is equivalent to the gyrocommutativity of the gyrogroup (G, +). 7) holds for all a, b∈G if and only if the gyrogroup (G, +) is ✷ gyrocommutative. 4. Let (G, +) be a gyrocommutative gyrogroup. 11) for all a, b, c ∈ G. Proof. 21, p. 38), p. 12) = gyr[a, b + c]gyr[b, c] . 1. 5. d = (b c) − a . 14) for all a, b, c ∈ G. Proof. 11), along with an application of the right and the left loop property, we have gyr[a , b + a ]gyr[b + a , c ] = gyr[a , b + c ]gyr[b + c , c ] . 17) =d.

34), p. 12. Follows from (3) by the gyration even property and by a left cancellation. 130). 130). Follows from (6) by the gyration even property. 34), p. 12. Follows from (8) by a right cancellation. 11. 35. 33 Let (G, +) be a gyrogroup. 144) of G into itself is bijective so that when a runs over all the elements of G its image, c, runs over all the elements of G as well for any given element b ∈ G. Proof. 144) is ✷ bijective and, hence, the result of the Corollary. 145) for all a, b, u, v ∈ G.