In Diestel graph theory book, theorem 6.3.3 (Tutte 1950) states:
A multigraph admits a $k$-flow iff it admits a $\mathbb{Z}_k$-flow.
I don't understand why do we need a proof, because, by definition, a $k$-flow is a flow all of whose values are between $1$ and $k$, and these are exactly the values we have in $\mathbb{Z}_k$ ... it looks to me that we called the set $\{1,2,\ldots,k-1\}$ by two names, say $A$ and $B$, and we claim that we have a flow with values from $A$ iff we have values from $B$....
What do I miss here?