The below-above germ pattern forces strict increase on the interval

Let $M$ be an o-minimal structure over a dense linear order without endpoints $D$, let $f : I \to B$ be a definable function of one variable, and let $u < v$ satisfy $(u,v) \subseteq I$.

Claim. If $f$ has the below-above germ pattern on $(u,v)$, then $f$ is strictly increasing on $(u,v)$:

The hypothesis only compares $f$ with the single value $f(x)$ immediately to the left and to the right of each point $x$; the conclusion is a genuinely global statement about all pairs of points of the interval. The passage from one to the other is the standard definable-connectedness argument: for fixed $x$ the set $\{z : f(z) > f(x)\}$ is definable, so it has finitely many boundary points, and the right half of the germ pattern produces a point of this set immediately to the right of any point already reached, while the left half shows that the set is inherited at a point from the points immediately to its left. Hence the set contains every point of $(u,v)$ above $x$.

Formalization note. UniformBelowAbove is the predicate of the accompanying definitions and IncOnInterval the strict-increase predicate of the mission's window-loci definitions.