Sum Less Maximum is Minimum

Theorem
For all numbers $$a, b$$ where $$a, b$$ in $$\mathbb{Z}, \mathbb{Q}$$ or $$\mathbb{R}$$:

$$a + b - \max \left\{{a, b}\right\} = \min \left\{{a, b}\right\}$$

Proof
This follows because subtracting the larger of $$a$$ and $$b$$ from their sum leaves the smaller.

Note that this does not apply to $$a, b \in \mathbb{C}$$ as there is no concept of ordering on the complex numbers $$\mathbb{C}$$.

Neither does it apply to $$a, b \in \mathbb{N}$$, as subtraction is not defined for all $$a$$ and $$b$$ in the natural numbers $$\mathbb{N}$$