Proof
(related to Proposition: Creation of Reduced Residue Systems From Others)
 By hypothesis, $m > 0$ is a positive integer, $n\perp m$ are coprime, and $R=\{a_1,\ldots,a_{\phi(m)}\}$ is a reduced residue system modulo $m$.
 Each of the $a_i$, $i=1,\ldots,\phi(m)$ are coprime to $m$, by definition of reduced residue systems, ($\phi(m)$ being the Euler function).
 Therefore, also $na_i$, $i=1,\ldots,\phi(m)$ are coprime to $m$, for $i=1,\ldots,\phi(m)$, by definition of coprimality.
 By the creation of complete residue systems from others, all are pairwise not congruent, i.e. $a_i(m)\not\equiv a_j(m)$, if and only if $i\neq j.$
 That means that $nR:=\{na_1,\ldots,na_{\phi(m)}\}$ is a reduced residue system modulo $m.$
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References
Bibliography
 Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927