# Real Number Ordering is Compatible with Addition

## Theorem

$\forall a, b, c \in \R: a < b \implies a + c < b + c$

where $\R$ is the set of real numbers.

## Proof

From Real Numbers form Ordered Integral Domain, $\struct {\R, +, \times, \le}$ forms an ordered integral domain.

By definition of ordered integral domain, the usual ordering $\le$ is compatible with ring addition.

$\blacksquare$

## Examples

### $15 + 3$ Greater than $12 + 3$

We have that:

$15 > 12$
$15 + 3 > 12 + 3$

That is:

$18 > 15$

### $15 - 3$ Greater than $12 - 3$

We have that:

$15 > 12$
$15 - 3 > 12 - 3$

That is:

$12 > 9$