| $P\ v\ Q$ |
| $R \rightarrow \sim P$ |
| $R\ \&\ S$ |
| $Q \rightarrow (R\ \&\ P)$ |
| $\therefore P\ v\ R$ |
| $(1)\ P\ v\ Q$ | $P$ |
| $(2)\ R\rightarrow \sim P$ | $P$ |
| $(3)\ R\ \&\ S$ | $P$ |
| $(4)\ Q\rightarrow (R\ \&\ P)$ | $P/\therefore P\ v\ R$ |
| $(5)\ R$ | $3,$ Simp. |
| $(6)\ \sim P$ | $2, 5, MP$ |
| $(7)\ Q$ | $1, 6, DS$ |
| $(8)\ R\ \&\ P$ | $4, 7, MP$ |
| $(9)\ P$ | $8,$ Simp. |
| $(10)\ P\ v\ R$ | $9,$ Add. |
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| $(R\ \&\ S)\ v\ P$ |
| $P\ \rightarrow\ Q$ |
| $T\ v\ (R\ \&\ S)\ \rightarrow\ W ]$ |
| $S\ \&\ \sim\ T$ |
| $[S\ \&\ (W\ v\ Q)]\ v\ \sim\ P$ |
| $N \ \&\ M$ |
| $\sim S\ (T \ \&\ W)$ |
| $(P \ \&\ R)\ v\ \sim S$ |
| $(N \ \&\ M) \rightarrow \sim (T \ \&\ W)$ |
| $(R \ \&\ N)\ v\ S$ |
| $(A\ v\ D) \rightarrow (R\ \&\ S)$ |
| $E\ v\ (A\ v\ D)$ |
| $F\ \&\ \sim E$ |
| $(R\ \&\ F)\ v\ K$ |
| $R\ \rightarrow\ (A\ \&\ B)$ |
| $P\ v\ \sim\ (S\ \&\ T)$ |
| $\sim\ T\ \&\ \sim\ P$ |
| $(A\ \&\ B)\ \rightarrow\ (S\ \&\ T)$ |
| $(\sim\ R\ \&\ \sim\ T)\ v\ D$ |
| $(A\ v\ B)\ \rightarrow\ [D\ \rightarrow\ (P\ \&\ \sim\ Q)]$ |
| $(A\ \&\ J)\ \rightarrow [(P\ \&\ \sim\ Q)\ \rightarrow\ K]$ |
| $(A\ \&\ J)\ \&\ (\sim\ K\ v\ D)$ |
| $\therefore\ (D \rightarrow\ K)\ v\ \sim\ Q$ |