There's a line in that proof saying r=min{a,b}; normally I take that to mean r is the minimum of a and b (which makes the proof wrong, since not all metric spaces have obvious orderings on their elements. Spaces like the complex plane or the 2d plane, with an appropriate metric, for instance.
I suppose it could mean r is the point in {a,b} such that the ball B_a(x) or B_b(x) has the smallest radius - but that looks more like a human making a notational mistake, particularly given that both 1b) and 1c) use min(,) in a way that seems correct to me (since they're using min on the values of the metric, not the elements of the metric space.
AFAICT, either the prover made a mistake in logic or a mistake with notation - which I reckon makes him or her human.
Then again, it's been years and years since I thought about this stuff, and I was prone to making mistakes all the time when I did, so everything above might well be wrong. My neurons are getting all fuzzy these days.