= Solution
<Uhlmann's theorem> identifies <quantum fidelity> with the largest absolute overlap of <purifications of a density operator>. Choose a common reference <Hilbert space> $R$ of dimension at least that of the original system. Then
$$
F(\rho_A,\sigma_A)=\max_{|u\rangle,|v\rangle}|\langle u|v\rangle|,
$$
where $|u\rangle,|v\rangle\in\mathcal H_A\otimes\mathcal H_R$ have reduced <density operators> $\rho_A,\sigma_A$. One purification may be fixed in advance: the maximum is over the other, with the freedom to apply a unitary on the reference system. This is the <unitary freedom of purification>. Enlarging the reference by unused dimensions does not change the maximum.
Take maximizing purifications $|u\rangle,|v\rangle$ of $\rho_{AB},\sigma_{AB}$ on $ABR$. Their overlap has magnitude $F(\rho_{AB},\sigma_{AB})$. Regard these same vectors as purifications of $\rho_A,\sigma_A$ with reference system $BR$. They are candidates in the larger optimization, so <Uhlmann's theorem> gives
$$
\boxed{F(\rho_A,\sigma_A)\geq F(\rho_{AB},\sigma_{AB}).}
$$
This is <monotonicity of quantum fidelity under partial trace>: discarding a system cannot make two states more distinguishable according to their <quantum fidelity>.
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