Multiple of Metric forms Metric

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

Let $d_1: A^2 \to \R$ be the mapping defined as:
 * $\forall \tuple {x, y} \in A^2: \map {d_1} {x, y} = k \map d {x, y}$

for some strictly positive $k \in \R_{>0}$.

Then $d_1$ is a metric for $A$.

Proof
It is to be demonstrated that $d_1$ satisfies all the metric space axioms.

Proof of
So holds for $d_1$.

Proof of
So holds for $d_1$.

Proof of
So holds for $d_1$.

Proof of
So holds for $d_1$.

Thus $d_1$ satisfies all the metric space axioms and so is a metric.