Start from the geostrophic and hydrostatic approximations $$ fv = \frac{1}{\rho_s r_0 \cos\theta}\frac{\partial p}{\partial \phi}, \qquad fu = - \frac{1}{\rho_s r_0}\frac{\partial p}{\partial \theta}, \qquad \rho g = - \frac{\partial p}{\partial z} $$

$$ fv = \frac{\dfrac{1}{\rho_s r_0 \cos\theta}\dfrac{\partial p}{\partial \phi}}{-\dfrac{1}{\rho}\dfrac{\partial p}{\partial z}}(-\rho g) \Rightarrow fv = -\frac{\rho g}{\rho_s r_0 \cos\theta}\frac{\dfrac{\partial p}{\partial \phi}}{\dfrac{\partial p}{\partial z}} = \frac{\rho}{\rho_s}\frac{g}{r_0 \cos\theta}\left(\frac{\partial z}{\partial \phi}\right)_p = \frac{g}{r_0 \cos\theta}\left(\frac{\partial z}{\partial \phi}\right)_p $$ Differentiate with respect to $z$ and using hydrostatic balance \[ f \frac{\partial v}{\partial z} = -\frac{g}{\rho_s r_0 \cos\theta}\frac{\partial \rho}{\partial \phi} - \frac{1}{\rho_s}\frac{\partial \rho_s}{\partial z}\, fv = -\frac{g}{\rho_s r_0 \cos\theta}\frac{\partial \rho}{\partial \phi} - \frac{g}{\rho_s r_0 \cos\theta}\left(\frac{\partial z}{\partial \phi}\right)_p \frac{\partial \rho_s}{\partial z} \] \[= -\frac{g}{\rho_s r_0 \cos\theta}\left[ \frac{\partial \rho}{\partial \phi} + \left(\frac{\partial z}{\partial \phi}\right)_p \frac{\partial \rho_s}{\partial z} \right]\] Replace $\rho_s$ by $\rho$ (to $O(\varepsilon)$), since $\rho = \rho_s + \rho'$, and $\rho' \ll \rho_s$, we can safely replace $\rho_s$ by $\rho$ in the denominator \[ f \frac{\partial v}{\partial z} = -\frac{g}{\rho r_0 \cos\theta} \left[ \frac{\partial \rho}{\partial \phi} + \frac{\partial \rho}{\partial z}\left(\frac{\partial z}{\partial \phi}\right)_p \right] \Rightarrow \frac{\partial v}{\partial z} = -\frac{g}{f \rho r_0 \cos\theta} \left(\frac{\partial \rho}{\partial \phi}\right)_p \] Repeat for the $u$-component $$ \frac{\partial u}{\partial z} = \frac{g}{f \rho r_0} \left(\frac{\partial \rho}{\partial \theta}\right)_p $$ In vector form $$ \frac{\partial \mathbf{u}_H}{\partial z} = - \frac{g}{\rho f}\, \mathbf{k} \times \left(\nabla \rho \right)_p $$ The increase of the horizontal velocity with height depends on the horizontal gradient of the density within surfaces of constant pressure. If the density and pressure surfaces are coincident, the geostrophic velocity must be independent of height. The geostrophic approximation therefore implies the Taylor–Proudman theorem.