Integer Addition is Well-Defined/Proof 2

Proof
Consider the formal definition of the integers: $x = \eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.

Consider the mapping $\phi: \N_{>0} \to \Z_{>0}$ defined as:
 * $\forall u \in \N_{>0}: \map \phi u = u'$

where $u' \in \Z$ be the (strictly) positive integer $\eqclass {b + u, b} {}$.

Let $v' = \eqclass {c + v, c} {}$.

Then: