= Solution
Apply the assumed <data-processing inequality for quantum relative entropy> to the normalized <partial trace> over $C$, with the two input states
$$
\rho_{ABC},
\qquad
\frac{I_A}{d_A}\otimes\rho_{BC}.
$$
The channel sends them to $\rho_{AB}\otimes I_C/d_C$ and $I_A/d_A\otimes\rho_B\otimes I_C/d_C$. Additivity over the common maximally mixed factor reduces data processing to
$$
D\left(\rho_{ABC}\middle\|\frac{I_A}{d_A}\otimes\rho_{BC}\right)
\geq
D\left(\rho_{AB}\middle\|\frac{I_A}{d_A}\otimes\rho_B\right).
$$
Expanding the <Umegaki relative entropy> in terms of <Von Neumann entropy> gives
$$
-S(\rho_{ABC})+\log d_A+S(\rho_{BC})
\geq
-S(\rho_{AB})+\log d_A+S(\rho_B).
$$
After cancelling $\log d_A$, this is precisely the <Strong subadditivity of Von Neumann entropy>
$$
\boxed{S(\rho_B)+S(\rho_{ABC})
\leq S(\rho_{AB})+S(\rho_{BC})}.
$$
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