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.