= Solution
The two <postulates of special relativity> are: \b[all nongravitational laws have the same form in every <inertial frame>], and \b[light in vacuum has the same finite <speed of light> $c$ in every <inertial frame>, independently of the motion of its source]. The first is <Einstein's principle of relativity>; the second supplies an experimentally identified invariant speed.
For the stated standard configuration, the <Lorentz transformation> is
$$
x'=\gamma_v(x-vt),\qquad t'=\gamma_v\left(t-\frac{vx}{c^2}\right),\qquad y'=y,\qquad z'=z,
\qquad \gamma_v=\frac1{\sqrt{1-v^2/c^2}}.
$$
Take <differentials> along the particle's <worldline>, rather than comparing coordinate displacements at equal times in the two frames. Writing $u_i=dx^i/dt$, one obtains
$$
dt'=\gamma_v\left(1-\frac{vu_x}{c^2}\right)dt,\qquad
dx'=\gamma_v(u_x-v)dt,\qquad dy'=u_y\,dt,\qquad dz'=u_z\,dt.
$$
Dividing by the transformed time interval proves the <velocity-addition formula>:
$$
\boxed{u'_x=\frac{u_x-v}{1-vu_x/c^2},\quad
u'_y=\frac{u_y}{\gamma_v(1-vu_x/c^2)},\quad
u'_z=\frac{u_z}{\gamma_v(1-vu_x/c^2)}.}
$$
For a massive particle, $|\mathbf u|<c$ and $|v|<c$, so the denominator $1-vu_x/c^2$ is positive.
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