| $R \rightarrow S$ |
| $P \rightarrow Q$ |
| $P\ v\ Q$ |
| $Q \rightarrow R$ |
| $\therefore (P \rightarrow R) \& (R\ v\ S)$ |
| $(1)\ R \rightarrow S$ | $P$ |
| $(2)\ P \rightarrow Q$ | $P$ |
| $(3)\ P\ v\ Q$ | $P$ |
| $(4)\ Q \rightarrow R$ | $P/\therefore\ (P\ \rightarrow\ R)\ \&\ (R\ v\ S)$ |
| $(5)\ P \rightarrow R$ | $2, 4, HS$ |
| $(6)\ Q \rightarrow S$ | $4, 1, HS$ |
| $(7)\ R\ v\ S$ | $5, 6, 3, CD$ |
| $(8)\ (P \rightarrow R)\ \&\ (R\ v\ S)$ | $5, 7,$ Conj. |
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| $(P \rightarrow Q)\ \&\ (R \rightarrow S)$ |
| $(Q \rightarrow T)\ \&\ (S \rightarrow P)$ |
| $\sim T$ |
| $\therefore \sim R\ \&\ \sim T$ |
| $A\ \&\ B.$ |
| $B\rightarrow (D\ v\ E)$ |
| $\sim E$ |
| $D \rightarrow (P \rightarrow \sim Q)$ |
| $\therefore P\rightarrow \sim Q$ |