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.