Proposition 11

Let E be an oriented plane over an ordered field A, D₀ a half-line of E, and G the set of half-lines of E distinct from D₀. The relation

“D₁ = D₂, or the sequence D₀, D₁, D₂ is direct”

between elements D₁, D₂ of G is a total ordering relation in G.

Indeed the axioms for total ordering relations are verified trivially, with the exception of transitivity. Let D₁, D₂, D₃ be three half-lines such that the sequences D₀, D₁, D₂ and D₀, D₂, D₃ are direct; we shall now prove that the sequence D₀, D₁, D₃ is direct. For this let us take a vector xᵢ ≠ 0 in Dᵢ (i = 0, 1, 2, 3), choose a bivector e > 0, and set xᵢ ∧ xⱼ = aᵢⱼe (aᵢⱼ ∈ A). Writing e = x₀ ∧ y (y ∈ E), and taking (x₀, y) as a basis of E, one easily verifies the relation

This being so, if a₀₁ ≤ 0, one has a₁₂ > 0 and a₂₀ > 0 (since the first sequence is direct), then a₂₃ > 0 and a₃₀ > 0 (since a₀₂ < 0 and the second sequence is direct), whence a₁₃ > 0 (by virtue of (22)); therefore the sequence (D₀, D₁, D₃) is direct in this case. Suppose henceforth that a₀₁ > 0. If a₃₀ ≤ 0, one has a₀₂ > 0 and a₂₃ > 0 (since the second sequence is direct), then a₁₂ > 0 (since a₂₀ < 0 and the first sequence is direct), whence $a_{13} > 0$ (by (22)), and the sequence $(D_0, D_1, D_3)$ is direct. Finally it is obviously analogous if $a_{01} > 0$ and $a_{30} > 0$. QED.