| $N \ \&\ M$ |
| $\sim S\ (T \ \&\ W)$ |
| $(P \ \&\ R)\ v\ \sim S$ |
| $(N \ \&\ M) \rightarrow \sim (T \ \&\ W)$ |
| $(R \ \&\ N)\ v\ S$ |
| $(1)\ N \ \&\ M$ | $P$ |
| $(2)\ \sim S \rightarrow (T \ \&\ W)$ | $P$ |
| $(3)\ (P \ \&\ R) v \sim S$ | $P$ |
| $(4)\ (N \ \&\ M)\ \rightarrow \sim (T \ \&\ W)$ | $P/ (R \ \&\ N)\ v\ S$ |
| $(5)\ \sim (T \ \&\ W)$ | $4, 1, MP$ |
| $(6)\ \sim\ \sim S$ | $2, 5, MT$ |
| $(7)\ P \ \&\ R$ | $3, 6, DS$ |
| $(8)\ R$ | $7,$ Simp. |
| $(9)\ N$ | $1,$ Simp. |
| $(10)\ R \ \&\ N$ | $8, 9,$ Conj. |
| $(11)\ (R \ \&\ N)\ v\ S$ | $10,$ Add. |
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| $R\ \rightarrow\ (S\ v\ T)$ |
| $P\ v((S\ v\ T)\ \rightarrow\ W]$ |
| $M\ v\ \sim\ P$ |
| $(H\ \&\ N)\ \rightarrow\ \sim\ M$ |
| $H\ \&\ N$ |
| $(R\ \rightarrow\ W)\ v\ S$ |
| $(T\ \&\ P) \rightarrow \sim Q$ |
| $\sim (T\ \&\ P) \rightarrow (R \rightarrow \sim Q)$ |
| $(\sim S\ v\ R)\rightarrow\ \sim\ \sim Q$ |
| $\sim S$ |
| $\therefore \sim R\ \&\ \sim S$ |