Direct proof that W ∪ parity is positional for ω-regular positional W

Develop a direct proof (not relying on the union-closure result and its machinery) that, for any ω-regular positional objective W, the objective W ∪ parity is positional.

Background

In their second decision procedure, the authors rely on the fact that W ∪ parity is positional for ω-regular positional W, which they derive from their main results. They point out the lack of a simpler, direct proof, whose existence would simplify the approach.