How should I prove 9.8? Could you include step by step explanation?

Theorem 9.8. A metric space is Hausdorff, regular, and normal. Definition. A metric on a set M is a function d : M X M - R+ (where R, is the non-negative real numbers) such that for all a, b, c E M, these properties hold: (1) d(a, b) 2 0, with d(a, b) = 0 if and only if a = b; (2) d(a, b) = d(b, a); (3) d(a, c) 0} forms a basis for a topology on X. The topology generated by a metric d on X is called the d-metric topology for X. Definition. A topological space (X, J) is a metric space or is metrizable if and only if there is a metric d on X such that J is the d-metric topology. We sometimes write a metric space as (X, d) to denote X with the d-metric topology. Theorem 9.6. For any metric space (X, d), there exists a metric d such that d and d generate the same topology, yet for each x, y E X, d(x, y)