Ordering on Cuts is Compatible with Addition of Cuts/Corollary

Theorem
Let $\alpha$ and $\gamma$ be cuts.

Let the operation of $\alpha + \gamma$ be the sum of $\alpha$ and $\gamma$. Let $0^*$ denote the rational cut associated with the (rational) number $0$.

If:
 * $\alpha > 0^*$ and $\gamma > 0^*$

then:
 * $\alpha + \gamma > 0^*$

where $>$ denotes the strict ordering on cuts.

Proof
From Ordering on Cuts is Compatible with Addition of Cuts


 * $0^* + 0^* < 0^* + \alpha$


 * $\alpha + 0^* < \alpha + \gamma$

The result follows from Ordering on Cuts is Transitive.