Uniqueness of Real z such that x Choose n+1 Equals y Choose n+1 Plus z Choose n

Theorem
Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $x, y \in \R$ be real numbers which satisfy:
 * $n \le y \le x \le y + 1$

Then there exists a unique real number $z$ such that:


 * $\dbinom x {n + 1} = \dbinom y {n + 1} + \dbinom z n$

where $n - 1 \le z \le y$.

Proof
We have: