| $(P\ \&\ Q) \rightarrow R$ |
| $R \rightarrow (H \rightarrow G)$ |
| $H\ \&\ K$ |
| $P\ \&\ Q$ |
| $\therefore (G\ v\ I)\ \&\ H$ |
| $(1)\ (P\ \&\ Q) \rightarrow R$ | $P$ |
| $(2)\ R \rightarrow (H \rightarrow G)$ | $P$ |
| $(3)\ H\ \&\ K$ | $P$ |
| $(4)\ P\ \&\ Q$ | $P/ \therefore (G\ v\ I)\ \&\ H$ |
| $(5)\ R$ | $1, 4, MP$ |
| $(6)\ H \rightarrow G$ | $2, 5, MP$ |
| $(7)\ H$ | $3,$ Simp. |
| $(8)\ G$ | $6, 7, MP$ |
| $(9)\ G\ v\ I$ | $8.$ Add. |
| $(10)\ (G\ \&\ I)$ | $9, 7,$ Conj. |
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| $A \rightarrow B$ |
| $(A\ \&\ B) \rightarrow C$ |
| $A$ |
| $\therefore (A\ \&\ B)\ \&\ (A\ \&\ C)$ |
| $J \rightarrow K$ |
| $J\ v\ (K\ v\ \sim\ L)$ |
| $\sim K$ |
| $\therefore \sim L\ \&\ \sim K$ |
| $H \rightarrow ( I\ \&\ \sim J)$ |
| $( I\ v\ G) \rightarrow K$ |
| $H$ |
| $\sim K\ \&\ I$ |
| $R \rightarrow S$ |
| $P \rightarrow Q$ |
| $P\ v\ Q$ |
| $Q \rightarrow R$ |
| $\therefore (P \rightarrow R) \& (R\ v\ S)$ |