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We prove the following theorem: Given a⊆ω and $1\le \alpha <{\omega }_1^{CK}$, if for some $\eta <{\aleph }_1$ and all u ∈ WO of length η, a is ${\Sigma }_{\alpha }^0\left(u\right)$, then a is ${\Sigma }_{\alpha }^0$. We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: ${\Sigma }_1^1$-Turing-determinacy implies the existence of $0$.