= Solution
The assertion is \b[true] over the complex numbers. It suffices to generate every nonzero scaling using translations and $I(z)=1/z$, since these together with scalings already generate the <Möbius group>.
For $s\ne0$, compose the maps rightmost first. Away from the intermediate poles, the successive values are
$$
z\longmapsto\frac1z\longmapsto\frac{1+sz}{z}\longmapsto\frac z{1+sz}
\longmapsto-\frac1{s(1+sz)}\longmapsto-s(1+sz)\longmapsto-s^2z.
$$
Hence
$$
\boxed{T_s\circ I\circ T_{-1/s}\circ I\circ T_s\circ I=D_{-s^2}.}
$$
For every $\lambda\ne0$, choose a complex square root $s$ with $s^2=-\lambda$. This gives $D_\lambda$ and proves <Möbius scaling generated by translations and reciprocal inversion>. Equality of the rational maps extends the identity to all points of the <Riemann sphere>, including the intermediate exceptional values.
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