| $(P\ v\ Q)\ \rightarrow\ (T\ v\ S)$ |
| $\sim\ N\ \& \sim\ M$ |
| $N\ v\ \sim\ (T\ v\ S)$ |
| $H\ \rightarrow\ (P\ v\ Q)$ |
| $\therefore \sim\ H\ \&\ \sim\ M$ |
| $(1)\ ( P\ v\ Q)\ \rightarrow\ (T\ v\ S)$ | $P$ |
| $(2)\ \sim\ N\ \&\ \sim\ M$ | $P$ |
| $(3)\ N\ v\ \sim\ (T\ v\ S)$ | $P$ |
| $(4)\ H\ \rightarrow\ (P\ v\ Q)$ | $P/ \therefore\ \sim\ H\ \&\ \rightarrow\ M$ |
| $(5)\ \sim\ N$ | $2,$ Simp. |
| $(6)\ \sim\ (T\ v\ S)$ | $3, 5, DS$ |
| $(7)\ \sim\ (P\ v\ Q)$ | $1, 6, MT$ |
| $(8)\ \sim\ H$ | $4, 7, MT$ |
| $(9)\ \sim\ M$ | $2,$ Simp. |
| $(10)\ \sim\ H\ \&\ \sim\ M$ | $8, 9,$ Conj. |
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| $A \rightarrow B$ |
| $B \rightarrow S$ |
| $A\ \&\ T$ |
| $\therefore S\ \&\ T$ |
| $(H\ \&\ K)\ \rightarrow\ (J\ v\ K)$ |
| $\sim\ E\ \&\ \sim\ F$ |
| $F\ v\ \sim\ (J\ v\ K)$ |
| $\sim\ (H\ \&\ K)\ \rightarrow\ H$ |
| $H\ \&\ \sim\ E$ |
| $H \rightarrow I$ |
| $H\ \&\ J$ |
| $I \rightarrow G$ |
| $G\ \&\ J$ |
| $\sim P\ \&\ (Q\ v\ R)$ |
| $(Q \rightarrow A)\ \&\ (R \rightarrow B)$ |
| $(A\ v\ B) \rightarrow (P\ v\ R)$ |
| $R$ |
| $E\ v\ (E\ \&\ Q)$ |
| $E \rightarrow L$ |
| $\sim L$ |
| $Q\ v\ M$ |
| $P\ v\ Q$ |
| $R \rightarrow \sim P$ |
| $R\ \&\ S$ |
| $Q \rightarrow (R\ \&\ P)$ |
| $\therefore P\ v\ R$ |