Sign of a Product of Adjacent Transpositions

Let $n\in\mathbb{N}$, let $\sigma\in S_n$ be a permutation in the sense of Permutation of the Set $\{1,\dots,r\}$, and suppose that

where each $\tau_{r_j}$ is an adjacent transposition as in Every Permutation is a Product of Adjacent Transpositions. Then

where $\operatorname{sgn}(\sigma)$ is the sign from Sign of a Permutation.