On the Relationship Between ID^_
<p>We show that the theory ATR<sub>0</sub> is equivalent to a second-order generalization of the theory $\widehat{ID}_{<\omega}$. As a result, ATR<sub>0</sub> is conservative over $\widehat{ID}_{<\omega}$ for arithmetic sentences, though proofs in ATR<sub>0</sub> can be much shorter than their $\widehat{ID}_{<\omega}$ counterparts.</p>