Positive Multiple of Metric is Metric

Theorem
Let $M = \struct {A, d}$ be a metric space.

Let $k \in \R_{>0}$ be a (strictly) positive real number.

Let $d_k: A \times A \to \R$ be the function defined as:
 * $\forall \tuple {x, y} \in A: \map {d_k} {x, y} = k \cdot \map d {x, y}$

Then $M_k = \struct {A, d_k}$ is a metric space.

So holds for $d_k$.

So holds for $d_k$.

So holds for $d_k$.

So holds for $d_k$.