Assume that $c$ is algebraic over $b_1, \ldots, b_n$. (Those $b_i$ are algebraic over $X$). Then there are functions $\varphi (c, b_1, \ldots, b_n)$ and $\chi_i(b_i)$.
$$\exists y_1, \ldots, \exists y_n : \chi_1(y_1) \wedge \ldots \wedge \chi_n(y_n) \wedge \varphi(x, y_1, \ldots, y_n)$$
$c$ realizes this formula, but why is this formula bounded?
