 Proof: By Clause (8), either \(\theta\) is atomic or it was produced by one of clauses (2)–(7). Thus, the first symbol in \(\theta\) must be either a predicate letter, a term, a unary marker, or a left parenthesis. If the first symbol in \(\theta\) is a predicate letter or term, then \(\theta\) is atomic. In this case, \(\theta\) was not produced by any of (2)–(7), since all such formulas begin with something other than a predicate letter or term. If the first symbol in \(\theta\) is a negation sign “\(\neg\)”, then was \(\theta\) produced by clause (2), and not by any other clause (since the other clauses produce formulas that begin with either a quantifier or a left parenthesis). Similarly, if \(\theta\) begins with a universal quantifier, then it was produced by clause (6), and not by any other clause, and if \(\theta\) begins with an existential quantifier, then it was produced by clause (7), and not by any other clause. The only case left is where \(\theta\) begins with a left parenthesis. In this case, it must have been produced by one of (3)–(5), and not by any other clause. We only need to rule out the possibility that \(\theta\) was produced by more than one of (3)–(5). To take an example, suppose that \(\theta\) was produced by (3) and (4). Then \(\theta\) is \((\psi_1 \amp \psi_2)\) and \(\theta\) is also \((\psi_3 \vee \psi_4)\), where \(\psi_1, \psi_2, \psi_3\), and \(\psi_4\) are themselves formulas. That is, \((\psi_1 \amp \psi_2)\) is the very same formula as \((\psi_3 \vee \psi_4)\). By Theorem \(5, \psi_1\) cannot be a proper part of \(\psi_3\), nor can \(\psi_3\) be a proper part of \(\psi_1\). So \(\psi_1\) must be the same formula as \(\psi_3\). But then “\(\amp\)” must be the same symbol as “\(\vee\)”, and this contradicts the policy that all of the symbols are different. So \(\theta\) was not produced by both Clause (3) and Clause (4). Similar reasoning takes care of the other combinations.