= Solution
Write $D=f[x_0,\ldots,x_n]$ and let $D_i$ be the <divided difference> with $x_i$ omitted. By symmetry from part (a), reorder the full list so that its first node is $x_i$ and its last is $x_j$. The recurrence from part (b) then has first omitted-node term $D_i$ and second omitted-node term $D_j$, and gives
$$
\boxed{D=\frac{D_i-D_j}{x_j-x_i}}.
$$
This proves the requested identity for every pair $i\ne j$ and fixes its sign explicitly; it is one of the <omitted-node identities for divided differences>.
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