 UV-compression proof of the Kruskal-Katona theorem

If a uniform family is not an initial colexicographic segment, choose a nontrivial $UV$-compression with $\max U<\max V$ and with $|U|$ minimal. The minimal choice supplies the smaller-compression hypotheses of the Shadow lemma for UV-compressions, so the operation does not increase the lower shadow. It strictly decreases the integer weightbecause the largest element of $U\cup V$ belongs to $V$. Iteration therefore terminates. A terminal family must be an initial colexicographic segment: otherwise an earlier absent set and a later present set supply one more admissible compression. The shadow of that segment has the size in the Kruskal-Katona theorem, proving the theorem.