Let $G = (V,E)$ be a directed graph with source $s$ and sink $t$ and $s \neq t$. For each edge $e \in E$, we have $c(e) \in \Bbb N$.
After running Ford-Fulkerson algorithm a flow function $f$ returned a max flow.
Let $R_f$ be the residual network that represents the max-flow $f$.
I would like to prove that if $R_f$ has no cycles then there is no other max flow function. I tried assuming that $R_f$ has no cycle and assuming towards contradiction that another flow function exists but I was stuck trying to find a cycle, thus reaching the contradiction. Any help will be appreciated.