Decomposing a closed walk in a directed graph into cycles

Let's say I have a closed walk $W$ in a directed graph $D$. Represent $W$ as a list of arcs $(a_1, a_2, \dots, a_k)$. Since it's a closed walk, there may be repeated edges, but the source vertex of $a_1$ must equal the target vertex of $a_k$. My claim is that the multiset of arcs $W'=\{a_1, a_2, \dots, a_k\}$ can be partitioned into sets (not multisets) $W'_1, W'_2, \dots, W'_m$ such that the arcs in each $W'_i$ form a simple cycle.

There is a simple algorithm to output these cycles. Travel along $W$ until you visit a vertex $v$ for the second time. Output the arcs between the two occurrences of $v$ (which form a cycle). Repeat this process until the walk is empty.

I'm using this "walk decomposition" lemma in a paper, and it seems like something that should be easy to find in the literature, but I can't find it anywhere. Can anyone point me in the right direction?

The problem stated in the question is also presented in the exercise of the book.

If you are writing a research paper, due to the is simplicity of the solution, you may even skip the explanation.

StackExchange Computer Science Q#53493, answer score: 2