Proposition 8

Suppose that A is a maximal ordered field, and that the form $\Phi$ is positive. Let $D_1, D_2, D'_1, D'_2$ be four lines (resp. half-lines) with origin 0 of E. A necessary and sufficient condition that the angles $(\widehat{D_1, D_2})$ and $(\widehat{D'_1, D'_2})$ be equal is that the angles $(\widehat{D_1, D'_1})$ and $(\widehat{D_2, D'_2})$ be equal.

Let us prove the necessity of the stated condition. Let u be a direct similitude (resp. a rotation) such that $u(D_1) = D'_1$ and $u(D_2) = D'_2$. There exists, by cor. 3 of prop. 1, a direct similitude (resp. by the cor. of prop. 7, a rotation) v such that $v(D_1) = D_2$. Since the group $S^+$ (resp. $O^+$) is commutative, we have $D'_2 = u(v(D_1)) = v(u(D_1)) = v(D'_1)$, and this shows that $(\widehat{D_1, D'_1}) = (\widehat{D_2, D'_2})$. The sufficiency is deduced from the necessity by interchanging $D_2$ and $D'_1$.

It follows from prop. 8 that to every angle $(\widehat{D_1, D_2})$ of lines (resp. half-lines) with origin 0 of E, there is canonically associated a well determined element of $S^+/H$ (resp. $O^+$), namely the class mod. H of the direct similitudes $\varphi$ (resp. the rotation $\varphi$) such that $u(D_1) = D_2$ for any representative $(D_1, D_2)$ of the angle $(\widehat{D_1, D_2})$. Thus there has been defined a canonical bijection $h$ (resp. $h'$) of the set $\mathfrak{A}_0$ of angles of lines (resp. $\mathfrak{A}$ of angles of half-lines) onto $S^+/H$ (resp. $O^+$); in particular, for every $\varphi \in \mathfrak{A}$, one says that $h(\varphi)$ is the rotation of angle $\varphi$. We shall transport to $\mathfrak{A}_0$ and $\mathfrak{A}$, by means of $h^{-1}$ and of ${h'}^{-1}$, the structures of commutative groups of $S^+/H$ and of $O^+$, and we shall denote additively the groups $\mathfrak{A}_0$ and $\mathfrak{A}$ thus obtained. If $D, D', D''$ denote lines (resp. half-lines) with origin 0 of E, we have by definition

whence one deduces