 For the definitions of the theories in this theorem, see end of  section 5.1.  To obtain the results about theories in (iii) and (iv) it is somewhat easier to first reduce them to systems of the form \((\Pi^0_1{\Hy}\bCA)_{&lt;\rho}\) as the latter have a straightforward interpretation in \({{\bRA}}^*\). Reductions of \((\Delta^1_1{\Hy}\bCR)\) to \((\Pi^0_1{\Hy}\bCA)_{&lt;\omega^{\omega}}\) and of \((\Sigma^1_1{\Hy}\bAC)\) to \((\Pi^0_1{\Hy}\bCA)_{&lt;\varepsilon_0}\) are due to Feferman (1964) and Friedman (1970),  respectively.[25] 