This is indeed much clearer than the OP's version. In this one, the transitivity is obvious, because the article clearly explains that you may take out any hors

This is indeed much clearer than the OP's version. In this one, the transitivity is obvious, because the article clearly explains that you may take out any horse from n + 1 and note that the remaining n have the same color, then do that observation again with another one, and use the horses that never left the group as a transitive link. I agree this is much more worth looking at.