Real Number Ordering is Compatible with Multiplication

Theorem

 * $\forall a, b, c \in \R: a < b \land c > 0 \implies a c < b c$


 * $\forall a, b, c \in \R: a < b \land c < 0 \implies a c > b c$

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

Proof
This follows from Rational Numbers form Subfield of Real Numbers.

Note
This Theorem can also be taken as an axiom for the real numbers, see Definition:Real Number/Axioms (Axiom RO2). If the real numbers are defined using limits of Cauchy sequences, it should be proved.