Because f has degree at most R-1 and W is a sum of identity and single-swap matrices, expanding f(W) uses products of at most R-1 swaps. Repeated swaps cancel in pairs. Multiplying by the full swap V leaves the permutation that swaps every position except those in a set S of size at most R-1. Each such permutation has mismatches only inside S. Therefore, whenever a row-column pair has more than R-1 mismatches, every expanded summand is zero there and so is M. Cancellation can remove additional entries; it cannot create an entry where every summand is zero. This proves the second property required by the replacement-row construction.