Ordering Properties of Real Numbers

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Theorem

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:

$(a): \quad a \le b \iff a < b \lor a = b$ ($a$ is less than or equal to $b$)

$(b): \quad a \ge b \iff a > b \lor a = b$ ($a$ is greater than or equal to $b$).

The following also holds:

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

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

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

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