Let X and d be given.
Assume Hd: metric_on X d.
We will prove topology_on X (metric_topology X d).
An exact proof term for the current goal is (lemma_topology_from_basis X (famunion X (λx ⇒ {open_ball X d x r|rR, Rlt 0 r})) (open_balls_form_basis X d Hd)).