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.