In the context of Walrasian demand:
Suppose u is continuous, satisfies local nonsatiation, and is strictly quasi-concave, each $w(p, x)$ contains a single consumption bundle.
The proof I got from a textbook is:
Let $x $~$ y$ with $p^T x=p^T y =w$, $z=\alpha x + (1- \alpha) y$ with $0 < \alpha < 1$. $p^T z =w$ because of the convexity of budget set.
Note: I understand why this makes sense graphically.
If $x \neq y$, strict quasi-concavity implies $u(z) > u(x) = u(y)$, thus $z$ is preferred to $x$ and $y$, hence $x, y$ are not elements of Walrasian demand.
Otherwise $x = y = z$.
For case 1, how can one be sure that the $z$ is unique?