Seminorm Maps Zero Vector to Zero

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

Let $X$ be a vector space over $\struct {K, \norm {\,\cdot\,}_K}$.

Let $\mathbf 0_X$ be the zero vector of $X$.

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

Then $\map p {\mathbf 0_X} = 0$.

Proof
We have: