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