Symmetry of Invariant Metric on Vector Space

Theorem
Let $K$ be a field.

Let $X$ be a vector space over $K$.

Let $d$ be an invariant metric on $X$.

Then we have:
 * $\map d {x, y} = \map d {-x, -y}$

for each $x, y \in X$.

Proof
Let $x, y \in X$.

We have: