Let $K$ be a proper cone. I need to prove following properties:

- if $x \preceq_K y$ and $u \preceq_K v$, then $x+u \preceq_K y+v$
- if $x \preceq_K y$ and $y \preceq_K z$, then $x \preceq_K z$
- if $x \preceq_K y$ and $\alpha \geq 0$, then $\alpha x \preceq_K \alpha y$
- $x \preceq_K x$

I can comprehend all of these in a graphical way as elements being ordered component-wise. But that does not qualify as a proof. What is the proper way of proving such properties?