Reflection principle for simple symmetric random walk

Let $S_0=0$ be a simple symmetric random walk, let $T_a$ be its first hitting time of the positive integer $a$, and reverse every increment after $T_a$. The resulting path is again a simple symmetric random walk. Consequently,

The continuous analogue is the existing Reflection principle (Wiener process); its tail formula has no discrete one-unit lattice correction.

For $M_n={\max }_{0\le k\le n}{S}_k$ and a positive integer $a$,Exactly one term can be nonzero because $S_n$ has the parity of $n$.