These notes provide and introduction to and proof of Proposition 3.9, the Tovey Property.
Proposition 3.9: For each u ≠ 0 and pair (s, z) with s < z, the function D(v, u, s, z)/(v – u), where
is strictly decreasing in v so long as i) v > u, ii) g(s, u, γ) = g(s, v, γ) < -p, iii) g(x, u, γ) > -p for some point x ≥ z and iv) D(v, u, s, z) > 0. Similarly, for each pair (z, S) with z < S, the function D(v, u, z, S)/(v – u) is strictly increasing in u so long as i) v > u, ii) g(S, u, γ) = g(S, v, γ) > -p, iii) g(x, v, γ) < -p for some point x ≤ z and iv) D(v, u, z, S) > 0.
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