Reverse Triangle Inequality/Seminormed Vector Space

Theorem
Let $\struct {K, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_K$.

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

Let $p$ be a seminorm on $X$.

Then:
 * $\forall x, y \in X : \size {\map p x - \map p y} \le \map p {x - y}$

Proof
We have:

We also have:

Therefore:
 * $- \map p {x - y} \le \map p x - \map p y \le \map p {x - y}$

In view of, the claim is proved.