Lemma 2.18 -- the reflection principle on $\mathbb{Z}$

Fix a natural number $r \ge 0$ and integers $k, j$ with $k > 0$ and $j > 0$. Sample paths are functions $\omega$ from $\{0, 1, \dots, r-1\}$ to $\{\mathrm{true}, \mathrm{false}\}$, i.e. sequences of $r$ Boolean coin flips (there are exactly $2^r$ of them, and when $r = 0$ there is exactly one, the empty sequence). For such an $\omega$ and a natural number $t$, the custom walk position $\mathrm{srwPos}\,k\,\omega\,t$, here written $S_t(\omega)$, is defined as

so $S_0(\omega) = k$ (the empty sum), $S_t(\omega)$ adds the first $t$ steps for $t \le r$, and for every $t \ge r$ (in particular $t = r$) it equals $k$ plus the sum of all $r$ steps, the walk's final position. The theorem asserts the conjunction of two exact counting identities over this finite space of Boolean sequences. First: the number of sequences $\omega$ such that there exists a natural number $s$ with $s < r$ and $S_s(\omega) = 0$ (the walk visits $0$ at some time strictly before $r$; since $S_0 = k > 0$, the time $s = 0$ can never witness this, and when $r = 0$ the condition is vacuously false) and $S_r(\omega) = j$, equals the number of sequences $\omega$ such that $S_r(\omega) = -j$ — note that this right-hand count carries no condition about visiting $0$: it counts all sequences whose final position is $-j$. Second: the number of sequences $\omega$ such that there exists $s < r$ with $S_s(\omega) = 0$ and $S_r(\omega) > 0$ (final position strictly positive), equals the number of sequences $\omega$ with $S_r(\omega) < 0$ (final position strictly negative), again with no zero-visit requirement on the right-hand side. Both identities are equalities of cardinalities of subsets of the $2^r$ Boolean sequences, stated for the fixed starting point $k > 0$ and target $j > 0$; nothing is asserted about probabilities, about $k \le 0$ or $j \le 0$, or about times other than "some $s < r$" and the final time $r$.