Removing a redundant input symbol from a channel

Suppose one row of a finite discrete memoryless channel is a convex combination of other rows: $W(\cdot \mid x_0)={\sum }_j{\alpha }_{j}W(\cdot \mid x_j)$. Deleting input $x_0$ leaves the channel capacity unchanged. Given any input law with mass $t$ on $x_0$, move that mass to the other symbols in proportions ${\alpha }_j$. The output law stays the same. By the concavity of information entropy, $H(W(\cdot \mid x_0))\ge {\sum }_j{\alpha }_{j}H(W(\cdot \mid x_j))$, so this replacement cannot increase conditional entropy $H(Y\mid X)$ and cannot decrease mutual information. Taking maxima proves the claim; the smaller channel is also trivially a restriction of the original one. If the entropy inequality is strict, every positive use of the redundant row is strictly suboptimal compared with its redistribution.