Integer Multiplication is Well-Defined/Proof 2

Theorem

Consider the mapping $\phi: \N_{>0} \to \Z_{>0}$ defined as:

$\forall u \in \N_{>0}: \map \phi u = u'$

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

Let:

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

Then:

$\ds u' v'$

$=$

$\ds \eqclass {b + u, b} {} \times \eqclass {c + v, c} {}$

$\ds $

$=$

$\ds \eqclass {\paren {b + u} \paren {c + v} + b c, \paren {b + u} c + b \paren {c + v} } {}$

$\ds $

$=$

$\ds \eqclass {b c + b v + c u + u v + b c, b c + u c + b c + b v} {}$

$\ds $

$=$

$\ds \eqclass {b c + u v, b c} {}$

$\ds $

$=$

$\ds \eqclass {b + u v, b} {}$

$\ds $

$=$

$\ds \paren {u v}'$

$\blacksquare$