Consider the product of cycles $\pi = \pi_1 \cdot \pi_2 \cdot \ldots \cdot \pi_r = \mu_1 \cdot \mu_2 \cdot \ldots \cdot \mu_s$. Suppose $\pi_i = (i_1, i_2, \ldots)$. Since the cycles are disjoint(if the cycles weren’t disjoint we’d get more than two cycles with the same element), there’s a unique cycle, say, $\mu_i = (j_1, j_2, \ldots)$ such that that $i_1 = j_1$ (any element of a cycle can be moved up to the initial position if need be). Then $i_2 = \pi(i_1) = \mu(j_1) = i_2$. Similarly, $i_3 = j_3$. The other cycles are dealt with in the same way.
Does the above make sense? Is it important here to mention that cycles commute?
Seems fine to me, no need to state that disjoint cycles commute (which is easy to prove anyway).
I would only emphasize more on the fact that it is sufficient to prove that any cycle in $\{\pi_1, ..., \pi_r\}$ is in $\{\mu_1, ..., \mu_r\}$ because both sets play a symmetrical role.