(b)    D contains a sentence,

        "txy [[Q_itx Ù S_jtx] ® [Q_mt'x' Ù S₀ty ® S₀t'y Ù ... Ù S_rty ® S_rt'y]]

        There is S_q such that that sentence together with d_s, [Succ], and [Diff] implies

        Q_me^s+1e^p+1 Ù S_j1e^s+1e^p1 Ù ... Ù S_je^s+1e^p Ù S_qe^s+1e^p+1 Ù ... Ù S_jve^s+1e^pv

                        Ù "y[[y¹e^p1 Ù ... Ù y¹e^p Ù y¹e^p+1 Ù ... Ù y¹e^pv] ® S₀e^s+1y]

        which is an example of  d_s+1.

(c)    Finally, D contains a sentence,

        "txy [[Q_itx' Ù S_jtx'] ® [Q_mt'x Ù S₀ty ® S₀t'y Ù ... Ù S_rty ® S_rt'y]]

        Again, there is S_q such that that sentence together with d_s, [Succ], and [Diff] implies

        Q_me^s+1e^p-1 Ù S_j1e^s+1e^p1 Ù ... Ù S_je^s+1e^p-1 Ù S_qe^s+1e^p Ù ... Ù S_jve^s+1e^pv

                        Ù "y[[y¹e^p1 Ù ... Ù y¹e^p-1 Ù y¹e^p Ù ... Ù y¹e^pv] ® S₀e^s+1y]

        which is an example of  d_s+1.

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    Therefore in all cases D |^_ d_s+1

    Therefore for any s > 0, if TM hasn't halted before s, D |^_ d_s

    So D |^_ H

    So if TM(n) halts then D |^_ H

    So D |^_ H iff TM(n) halts

    So a solution to this problem is a solution to the halting problem

    So there is no solution to this decision problem.