Ordering Properties of Real Numbers

Trichotomy Law
The real numbers obey the Trichotomy Law. That is, $$\forall a, b \in \R$$, exactly one of the following holds:


 * $$(1) \quad a > b$$ ($$a$$ is greater than $$b$$);
 * $$(2) \quad a = b$$ ($$a$$ is equal to $$b$$);
 * $$(3) \quad a < b$$ ($$a$$ is less than $$b$$).

Note that $$a > b \iff b < a$$.

We also use the following notation:


 * $$(a) \quad a \le b \iff a < b \or a = b$$ ($$a$$ is less than or equal to $$b$$);
 * $$(b) \quad a \ge b \iff a > b \or a = b$$ ($$a$$ is greater than or equal to $$b$$).

The following also holds:


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

Ordering is Compatible with Addition

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

Ordering is Compatible with Multiplication

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


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

Proof of Trichotomy Law
This follows directly from the fact that the real numbers form a totally ordered field.

Proof that Ordering is Compatible with Addition
This follows from the fact that the Reals are an Extension of Rationals.

Proof that Ordering is Compatible with Multiplication
This follows from the fact that the Reals are an Extension of Rationals.