Exercise 3, § 3

Let $A_1$, $A_2$, ..., $A_n$ be relations in a logical theory $\mathscr{T}$.

(a) To prove the relation "$A_1$ or $A_2$ or ... or $A_n$" in $\mathscr{T}$, it is enough to prove $A_n$ in the theory obtained by adjoining to $\mathscr{T}$ the axioms not $A_1$, not $A_2$, ..., not $A_{n-1}$.

(b) If "$A_1$ or $A_2$ or ... or $A_n$" is a theorem in $\mathscr{T}$, then to prove a theorem $A$ in $\mathscr{T}$ it is sufficient to prove the theorems