Quantifier has Unique Scope

Theorem
Let $\mathbf A$ be a WFF of predicate calculus.

Let $Q$ be a quantifier that occurs in $\mathbf A$.

Then there is a unique well-formed part of $\mathbf A$ which begins with $Q$.

This unique well-formed part of $\mathbf A$ is called the scope of that occurrence of $Q$.

Proof
First, from the rules of formation of predicate calculus, we have that whenever a quantifier is included in a WFF, it appears in the form:
 * $Q x: \mathbf B$

where $\mathbf B$ is itself a WFF.

Hence it is clear that $Q x: \mathbf B$ is one well-formed part of $\mathbf A$ which begins with $Q$.

Now we prove that this well-formed part is unique.

Let $Q$ be an occurrence of a quantifier in a WFF $\mathbf A$.

Suppose $\mathbf B$ and $\mathbf C$ are both well-formed parts of $\mathbf A$ which begin with $Q$.

Since $\mathbf B$ and $\mathbf C$ both begin with $Q$, neither one can be the initial part of the other, as No Initial Part of a WFF of PredCalc is a WFF.

So $\mathbf B$ and $\mathbf C$ are the same.

Hence the result.