219 Deductive Reasoning in First-order Logic 4.6.9 Prove Lemma 166 for: (a) H; (b) ST; (c) ND; (d) RES 4.6.10 Prove Lemma 168 generically for any deductive system D 4.6.11 Prove Lemma 170 4.6.12 Complete the proof of Lemma 172 for: (a) H; (b) ST; (c) ND; (d) RES Prove Lemma 174 for: (a) H; (b) ST; (d) RES 4.6.13 (c) ND; (Hint for ND: suppose Γ D ⊥ in L+ by a derivation Ξ Let Ξ be the result of replacing each free occurrence of a new constant ci in Ξ by a new variable xi not occurring in Ξ It can be shown, by inspection of the rules of D, that Ξ is a valid derivation of Γ D ⊥ in L.) 4.6.14 Prove Lemma 175 for: (a) H; (b) ST; (c) ND; (d) RES 4.6.15 Prove that the deductive compactness property stated in Lemma 175 is equivalent to the following: a first-order theory Γ is D-consistent iff every finite subset of Γ is D-consistent 4.6.16 Complete the generic proof details of Theorem 176 for any deductive system D 4.6.17 Assuming soundness and completeness of H, prove soundness and completeness of each of ST, ND, and RES by using a first-order analog of Proposition 76 4.6.18 Assuming soundness and completeness of ST, prove soundness and completeness of each of H, ND, and RES by using a first-order analog of Proposition 76 4.6.19 Assuming soundness and completeness of ND, prove soundness and completeness of each of ST, H, and RES by using a first-order analog of Proposition 76 4.6.20 Assuming soundness and completeness of RES, prove soundness and completeness of each of H, ST, and ND by using a first-order analog of Proposition 76 4.6.21 Prove the equivalence of the semantic compactness theorems 179 and 180