Existence of Unique Difference between Cuts

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

Then there exists exactly one cut $\gamma$ such that:


 * $\alpha + \gamma = \beta$

Proof
From Ordering on Cuts is Compatible with Addition of Cuts:
 * $\gamma_1 \ne \gamma_2 \implies \alpha + \gamma_1 \ne \alpha + \gamma_2$

That demonstrates uniqueness.

Let:
 * $\gamma = \beta + \paren {-\alpha}$

where $-\alpha$ is the negative of $\alpha$.

Then by Identity Element for Addition of Cuts:

The result follows.

Also see

 * Definition:Subtraction of Cuts: $\gamma$ is defined as $\beta - \alpha$, the difference between $\beta$ and $\alpha$