| $P \rightarrow Q$ |
| $\sim Q\ v\ R$ |
| $\sim R$ |
| $\therefore (\sim P \& \sim R)\ v\ S$ |
| $(1)\ P \rightarrow Q$ | $P$ |
| $(2)\ \sim Q\ v\ R$ | $P$ |
| $(3)\ \sim R$ | $P/ \therefore (\sim P\ \&\ \sim R)\ v\ S$ |
| $(4)\ \sim Q$ | $2, 3, DS$ |
| $(5)\ \sim p$ | $1, 4, MT$ |
| $(6)\ \sim P\ \&\ \sim R$ | $5, 3,$ Conj. |
| $(7)\ (\sim P\ \&\ \sim R)\ v\ S$ | $6 $ Add. |
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| $G \rightarrow J$ |
| $J \rightarrow K$ |
| $(G \rightarrow K) v (J \rightarrow L)$ |
| $L \rightarrow M$ |
| $\therefore (J \rightarrow M) v\ Q$ |
| $T \rightarrow Q$ |
| $B \rightarrow (K\ v\ T)$ |
| $(N \rightarrow B) \rightarrow (K\ v\ B)$ |
| $K\ v\ (N \rightarrow B)$ |
| $\sim K$ |
| $\therefore\ \sim Q\ v\ Z$ |
| $A \rightarrow B$ |
| $C \rightarrow D$ |
| $\sim B\ \&\ \sim D$ |
| $(\sim A\ \&\ \sim C)\ v\ P$ |
| $R\ \rightarrow\ (A\ \&\ B)$ |
| $P\ v\ \sim\ (S\ \&\ T)$ |
| $\sim\ T\ \&\ \sim\ P$ |
| $(A\ \&\ B)\ \rightarrow\ (S\ \&\ T)$ |
| $(\sim\ R\ \&\ \sim\ T)\ v\ D$ |
| $A \rightarrow B$ |
| $C \rightarrow B$ |
| $(\sim\ A\ \&\ \sim \ C)\ \rightarrow\ (D\ \rightarrow\ E)$ |
| $(E\ \rightarrow\ G)\ \&\ (D\ v\ E)$ |
| $E\ v\ G$ |
| $N \rightarrow (\sim Z \rightarrow G)$ |
| $D \rightarrow [K\ v\ Y) \rightarrow M]$ |
| $(X\ v\ Y\ v\ \sim\ Z$ |
| $D\ \&\ N$ |
| $\therefore\ (M\ v\ G)\ v\ K$ |