$\forall$$m$, ${\it m'}$, $a$, ${\it a'}$, $b$, ${\it b'}$:$\mathbb{Z}$.
\\[0ex]$m$ $=$ ${\it m'}$ $\Rightarrow$ ($a$ = ${\it a'}$ mod $m$) $\Rightarrow$ ($b$ = ${\it b'}$ mod $m$) $\Rightarrow$ (($a$ = $b$ mod $m$) $\Leftrightarrow$ (${\it a'}$ = ${\it b'}$ mod ${\it m'}$))