= Solution
For any <injection> $u:U\hookrightarrow V$, the <direct sum> decomposition $F=P\oplus Q$ gives natural <module isomorphisms>
$$
U\otimes_AF\cong(U\otimes_AP)\oplus(U\otimes_AQ),\qquad V\otimes_AF\cong(V\otimes_AP)\oplus(V\otimes_AQ).
$$
Under these, $u\otimes1_F$ is the block-diagonal map $(u\otimes1_P)\oplus(u\otimes1_Q)$. It is <injective> because $F$ is a <flat module>. If an element lies in the <kernel> of $u\otimes1_P$, insert zero in the other component; injectivity of the whole map forces that element to be zero. Hence $u\otimes1_P$ is <injective> for every $u$, which proves $\boxed{P\text{ is flat}}$. This establishes that <direct summands of flat modules are flat>, without a finite-generation assumption.
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