Extra Credit $1($Craig Interpolation Theorem$)$: Let $$ and $$ be formulas

of the language of propositional logic. Suppose that $$ is satisfiable, $$ is

not a tautology, and $()$ is a tautology.

Your assignment: Show that there exists a formula $$ of the language

such that $($i$)$ every proposition symbol that occurs in $$ also occurs in both

$$ and $,$ and $($ii$)$ both $()$ and $()$ are tautologies. $($Hint $\{$:${?}^1)$