I have always see the following Murnaghan-Nakayama rule for a partition $\lambda$ and a permutation $\sigma \in \mathfrak{S}_n$ of cycle structure $(\sigma_1, ..., \sigma_n)$: $$ \chi_{\lambda}(\sigma) = \displaystyle \sum_{\xi \in R(\lambda, \sigma_1)} (-1)^{ht(\xi)} \chi_{\lambda \setminus \xi}(\sigma \setminus \sigma_1). $$ where $R(\lambda, \sigma_1)$ are the skew-hooks of $\lambda$ of length $\sigma_1$.

However, in the proof I am reading, they use a somewhat different rule. Let $\sigma \in \mathfrak{S}_{n+m}$ be a permutation that fixes (at least) $m$ elements and $\tau$ be a permutation of the $m$ elements fixed by $\sigma$. Then, $$ \chi_{\lambda}(\sigma \tau) = \displaystyle \sum_{\xi \in R(\lambda, \sigma_1)} (-1)^{ht(\xi)} \chi_{\lambda \setminus \xi}(\sigma). $$ The equivalence between the two formulas is clear if $|\tau| \geq \sigma_1$, but I don't see why this is true when $|\tau| < \sigma_1$. Is it trivial? Am I missing something, or it is a non-trivial corolary?

Thanks in advance!