Manuscript III@2 · III.tex · lemma · 82dad3e5

Lemma: Two-representative estimate

The unchanged free-product obstruction › A fixed nonzero gap · lines 738–750 · proof included or linked

III@2#lem:two-point

If $q (x_{0}) = q (x_{1})$ , then for $H : X \to C$ and $G : Y \to C$ ,

j = 0, 1 max ∣ G (q (x_{j})) - H (x_{j}) ∣ \geq \frac{1}{2} ∣ H (x_{1}) - H (x_{0}) ∣.

Proof. Insert the common value $G (q (x_{0})) = G (q (x_{1}))$ between the two values of $H$ , and use the triangle inequality.

Displayed formulas in this passage

unlabeled · line 741
$j = 0, 1 max ∣ G (q (x_{j})) - H (x_{j}) ∣ \geq \frac{1}{2} ∣ H (x_{1}) - H (x_{0}) ∣.$