Law of Inverses (Modulo Arithmetic)/Corollary 2

Corollary to Law of Inverses (Modulo Arithmetic)
Let $m, n \in \Z$ such that:
 * $m \perp n$

that is, such that $m$ and $n$ are coprime.

Then:
 * $\exists n' \in \Z: n n' \equiv 1 \pmod m$

where:
 * $n' \equiv n^{\map \phi n - 1} \pmod m$

where $\map \phi n$ is the Euler $\phi$ function.

Proof
From Law of Inverses (Modulo Arithmetic): Corollary 1:
 * $\exists n' \in \Z: n n' \equiv 1 \pmod m$

From Euler's Theorem:
 * $n^{\map \phi m} \equiv 1 \pmod m$

it follows that:
 * $n \cdot n^{\map \phi m - 1} \equiv 1 \pmod m$

The result follows directly.