 Letter-merging proof of the Hales-Jewett theorem

Induct on the alphabet size $k$. For an $r$-colouring, choose an $(k-1,k)$-insensitive combinatorial subspace whose dimension forces a monochromatic combinatorial line over the first $k-1$ letters. The inductive Hales-Jewett theorem supplies that line. Changing all of its active parameters from $k-1$ to $k$, one at a time, preserves the colour by the alphabet insensitivity lemma. Thus the line extends to all $k$ letters. The one-letter alphabet is the base case.