Ordering on Integers is Transitive

Theorem
Let $\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers.

Then:

That is, ordering on the integers is transitive.

Proof
By the formal definition of integers, we have that $a, b, c, d, e, f$ are all natural numbers.

To eliminate confusion between integer ordering and the ordering on the natural numbers, let $a \preccurlyeq b$ denote that the natural number $a$ is less than or equal to the natural number $b$.

We have: