I am currently reading this [paper](https://arxiv.org/pdf/1704.01224.pdf), where the author classifies the pretzel links up to link homotopy using a quasi-trivial quandle $\mathbb{Z}_{k}[t^{\pm 1}]\diagup_{(t-1)^{2}}$, and I find it difficult to understand the proof of lemma 5.6. Can anyone provide some help for me to understand this proof?