Positive Difference Relation on Reals is Transitive

Theorem
Let $P \subseteq \R$ be a subset of the real numbers such that:
 * $(1): \quad 1 \in P$
 * $(2): \quad a, b \in P \implies a + b \in P$
 * $(3): \quad$ For all $x \in \R$, exactly one of these is true:
 * $x \in P$
 * $x = 0$
 * $-x \in P$

Let $Q \subseteq \R \times \R$ be the relation on $\R$ defined as:


 * $Q = \set {\tuple {a, b} \in \R: a - b \in P}$

Then $Q$ is a transitive relation.

Proof
Let $a - b \in P$ and $b - c \in P$.

By condition $(2)$:
 * $\paren {a - b} + \paren {b - c} \in P$

Simplifying:
 * $a - c \in P$

The result follows by definition of transitive relation.