 We sketch a proof that \(\Gamma'\) is consistent. Suppose that \(\Gamma'\) is inconsistent. By Theorem 9, there is a finite subset of \(\Gamma\) that is inconsistent, and so one of the sets \(\Gamma_m\) is inconsistent. By hypothesis, \(\Gamma_0 = \Gamma\) is consistent. Let \(n\) be the smallest number such that \(\Gamma_n\) is consistent, but \(\Gamma_{n+1} = \Gamma_n,(\exists x\theta_n \rightarrow \theta_{n}(x|c_i))\) is inconsistent. By \((\neg\)I), we have that